diff --git a/src/Magnum/AbstractTexture.cpp b/src/Magnum/AbstractTexture.cpp
index 73804e907..a63d2ba05 100644
--- a/src/Magnum/AbstractTexture.cpp
+++ b/src/Magnum/AbstractTexture.cpp
@@ -1616,9 +1616,17 @@ template void MAGNUM_EXPORT AbstractTexture::image<3>(GLint, BufferImage<3>&, Bu
 
 template<UnsignedInt dimensions> void AbstractTexture::compressedImage(const GLint level, CompressedImage<dimensions>& image) {
     const Math::Vector<dimensions, Int> size = DataHelper<dimensions>::imageSize(*this, level);
-    GLint textureDataSize;
-    (this->*Context::current().state().texture->getLevelParameterivImplementation)(level, GL_TEXTURE_COMPRESSED_IMAGE_SIZE, &textureDataSize);
-    const std::size_t dataSize = Implementation::compressedImageDataSizeFor(image, size, textureDataSize);
+
+    /* If the user-provided pixel storage doesn't tell us all properties about
+       the compression, we need to ask GL for it */
+    std::size_t dataSize;
+    if(!image.storage().compressedBlockSize().product() || !image.storage().compressedBlockDataSize()) {
+        GLint textureDataSize;
+        (this->*Context::current().state().texture->getLevelParameterivImplementation)(level, GL_TEXTURE_COMPRESSED_IMAGE_SIZE, &textureDataSize);
+        dataSize = textureDataSize;
+    } else dataSize = Implementation::compressedImageDataSizeFor(image, size);
+
+    /* Internal texture format */
     GLint format;
     (this->*Context::current().state().texture->getLevelParameterivImplementation)(level, GL_TEXTURE_INTERNAL_FORMAT, &format);
 
@@ -1639,9 +1647,17 @@ template void MAGNUM_EXPORT AbstractTexture::compressedImage<3>(GLint, Compresse
 
 template<UnsignedInt dimensions> void AbstractTexture::compressedImage(const GLint level, CompressedBufferImage<dimensions>& image, BufferUsage usage) {
     const Math::Vector<dimensions, Int> size = DataHelper<dimensions>::imageSize(*this, level);
-    GLint textureDataSize;
-    (this->*Context::current().state().texture->getLevelParameterivImplementation)(level, GL_TEXTURE_COMPRESSED_IMAGE_SIZE, &textureDataSize);
-    const std::size_t dataSize = Implementation::compressedImageDataSizeFor(image, size, textureDataSize);
+
+    /* If the user-provided pixel storage doesn't tell us all properties about
+       the compression, we need to ask GL for it */
+    std::size_t dataSize;
+    if(!image.storage().compressedBlockSize().product() || !image.storage().compressedBlockDataSize()) {
+        GLint textureDataSize;
+        (this->*Context::current().state().texture->getLevelParameterivImplementation)(level, GL_TEXTURE_COMPRESSED_IMAGE_SIZE, &textureDataSize);
+        dataSize = textureDataSize;
+    } else dataSize = Implementation::compressedImageDataSizeFor(image, size);
+
+    /* Internal texture format */
     GLint format;
     (this->*Context::current().state().texture->getLevelParameterivImplementation)(level, GL_TEXTURE_INTERNAL_FORMAT, &format);
 
@@ -1720,9 +1736,18 @@ template<UnsignedInt dimensions> void AbstractTexture::compressedSubImage(const
     const Math::Vector<dimensions, Int> size = range.size();
     const Vector3i paddedOffset = Vector3i::pad(range.min());
     const Vector3i paddedSize = Vector3i::pad(size, 1);
+
+    /* Internal texture format */
     GLint format;
     (this->*Context::current().state().texture->getLevelParameterivImplementation)(level, GL_TEXTURE_INTERNAL_FORMAT, &format);
-    const std::size_t dataSize = Implementation::compressedImageDataSizeFor(image, size, compressedSubImageSize<dimensions>(TextureFormat(format), size));
+
+    /* Calculate compressed subimage size. If the user-provided pixel storage
+       doesn't tell us all properties about the compression, we need to ask GL
+       for it. That requires GL_ARB_internalformat_query2. */
+    std::size_t dataSize;
+    if(!image.storage().compressedBlockSize().product() || !image.storage().compressedBlockDataSize())
+        dataSize = compressedSubImageSize<dimensions>(TextureFormat(format), size);
+    else dataSize = Implementation::compressedImageDataSizeFor(image, size);
 
     /* Reallocate only if needed */
     Containers::Array<char> data{image.release()};
@@ -1745,9 +1770,18 @@ template<UnsignedInt dimensions> void AbstractTexture::compressedSubImage(const
     const Math::Vector<dimensions, Int> size = range.size();
     const Vector3i paddedOffset = Vector3i::pad(range.min());
     const Vector3i paddedSize = Vector3i::pad(size, 1);
+
+    /* Internal texture format */
     GLint format;
     (this->*Context::current().state().texture->getLevelParameterivImplementation)(level, GL_TEXTURE_INTERNAL_FORMAT, &format);
-    const std::size_t dataSize = Implementation::compressedImageDataSizeFor(image, size, compressedSubImageSize<dimensions>(TextureFormat(format), size));
+
+    /* Calculate compressed subimage size. If the user-provided pixel storage
+       doesn't tell us all properties about the compression, we need to ask GL
+       for it. That requires GL_ARB_internalformat_query2. */
+    std::size_t dataSize;
+    if(!image.storage().compressedBlockSize().product() || !image.storage().compressedBlockDataSize())
+        dataSize = compressedSubImageSize<dimensions>(TextureFormat(format), size);
+    else dataSize = Implementation::compressedImageDataSizeFor(image, size);
 
     /* Reallocate only if needed */
     if(image.dataSize() < dataSize)
diff --git a/src/Magnum/CubeMapTexture.cpp b/src/Magnum/CubeMapTexture.cpp
index bf5b3ee9a..351ecbe72 100644
--- a/src/Magnum/CubeMapTexture.cpp
+++ b/src/Magnum/CubeMapTexture.cpp
@@ -109,9 +109,16 @@ void CubeMapTexture::compressedImage(const Int level, CompressedImage3D& image)
     createIfNotAlready();
 
     const Vector3i size{imageSize(level), 6};
-    const GLint textureDataSize = (this->*Context::current().state().texture->getCubeLevelCompressedImageSizeImplementation)(level)*6;
+
+    /* If the user-provided pixel storage doesn't tell us all properties about
+       the compression, we need to ask GL for it */
     std::size_t dataOffset, dataSize;
-    std::tie(dataOffset, dataSize) = Implementation::compressedImageDataOffsetSizeFor(image, size, textureDataSize);
+    if(!image.storage().compressedBlockSize().product() || !image.storage().compressedBlockDataSize()) {
+        dataOffset = 0;
+        dataSize = (this->*Context::current().state().texture->getCubeLevelCompressedImageSizeImplementation)(level)*6;
+    } else std::tie(dataOffset, dataSize) = Implementation::compressedImageDataOffsetSizeFor(image, size);
+
+    /* Internal texture format */
     GLint format;
     (this->*Context::current().state().texture->getCubeLevelParameterivImplementation)(level, GL_TEXTURE_INTERNAL_FORMAT, &format);
 
@@ -135,9 +142,16 @@ void CubeMapTexture::compressedImage(const Int level, CompressedBufferImage3D& i
     createIfNotAlready();
 
     const Vector3i size{imageSize(level), 6};
-    const GLint textureDataSize = (this->*Context::current().state().texture->getCubeLevelCompressedImageSizeImplementation)(level)*6;
+
+    /* If the user-provided pixel storage doesn't tell us all properties about
+       the compression, we need to ask GL for it */
     std::size_t dataOffset, dataSize;
-    std::tie(dataOffset, dataSize) = Implementation::compressedImageDataOffsetSizeFor(image, size, textureDataSize);
+    if(!image.storage().compressedBlockSize().product() || !image.storage().compressedBlockDataSize()) {
+        dataOffset = 0;
+        dataSize = (this->*Context::current().state().texture->getCubeLevelCompressedImageSizeImplementation)(level)*6;
+    } else std::tie(dataOffset, dataSize) = Implementation::compressedImageDataOffsetSizeFor(image, size);
+
+    /* Internal texture format */
     GLint format;
     (this->*Context::current().state().texture->getCubeLevelParameterivImplementation)(level, GL_TEXTURE_INTERNAL_FORMAT, &format);
 
@@ -199,8 +213,16 @@ BufferImage2D CubeMapTexture::image(const CubeMapCoordinate coordinate, const In
 
 void CubeMapTexture::compressedImage(const CubeMapCoordinate coordinate, const Int level, CompressedImage2D& image) {
     const Vector2i size = imageSize(level);
-    const GLint textureDataSize = (this->*Context::current().state().texture->getCubeLevelCompressedImageSizeImplementation)(level);
-    const std::size_t dataSize = Implementation::compressedImageDataSizeFor(image, size, textureDataSize);
+
+    /* If the user-provided pixel storage doesn't tell us all properties about
+       the compression, we need to ask GL for it */
+    std::size_t dataSize;
+    if(!image.storage().compressedBlockSize().product() || !image.storage().compressedBlockDataSize())
+        dataSize = (this->*Context::current().state().texture->getCubeLevelCompressedImageSizeImplementation)(level);
+    else
+        dataSize = Implementation::compressedImageDataSizeFor(image, size);
+
+    /* Internal texture format */
     GLint format;
     (this->*Context::current().state().texture->getCubeLevelParameterivImplementation)(level, GL_TEXTURE_INTERNAL_FORMAT, &format);
 
@@ -222,8 +244,16 @@ CompressedImage2D CubeMapTexture::compressedImage(const CubeMapCoordinate coordi
 
 void CubeMapTexture::compressedImage(const CubeMapCoordinate coordinate, const Int level, CompressedBufferImage2D& image, const BufferUsage usage) {
     const Vector2i size = imageSize(level);
-    const GLint textureDataSize = (this->*Context::current().state().texture->getCubeLevelCompressedImageSizeImplementation)(level);
-    const std::size_t dataSize = Implementation::compressedImageDataSizeFor(image, size, textureDataSize);
+
+    /* If the user-provided pixel storage doesn't tell us all properties about
+       the compression, we need to ask GL for it */
+    std::size_t dataSize;
+    if(!image.storage().compressedBlockSize().product() || !image.storage().compressedBlockDataSize())
+        dataSize = (this->*Context::current().state().texture->getCubeLevelCompressedImageSizeImplementation)(level);
+    else
+        dataSize = Implementation::compressedImageDataSizeFor(image, size);
+
+    /* Internal texture format */
     GLint format;
     (this->*Context::current().state().texture->getCubeLevelParameterivImplementation)(level, GL_TEXTURE_INTERNAL_FORMAT, &format);
 
diff --git a/src/Magnum/CubeMapTexture.h b/src/Magnum/CubeMapTexture.h
index cb320546c..112e09e4c 100644
--- a/src/Magnum/CubeMapTexture.h
+++ b/src/Magnum/CubeMapTexture.h
@@ -792,6 +792,10 @@ class MAGNUM_EXPORT CubeMapTexture: public AbstractTexture {
          * See @ref Texture::compressedSubImage(Int, const RangeTypeFor<dimensions, Int>&, CompressedImage&)
          * for more information.
          * @requires_gl45 Extension @extension{ARB,get_texture_sub_image}
+         * @requires_gl43 Extension @extension{ARB,internalformat_query2} if
+         *      @ref CompressedPixelStorage::compressedBlockSize() and
+         *      @ref CompressedPixelStorage::compressedBlockDataSize() are not
+         *      set to non-zero values
          * @requires_gl Texture image queries are not available in OpenGL ES or
          *      WebGL. See @ref Framebuffer::read() for possible workaround.
          */
@@ -814,6 +818,10 @@ class MAGNUM_EXPORT CubeMapTexture: public AbstractTexture {
          * See @ref Texture::compressedSubImage(Int, const RangeTypeFor<dimensions, Int>&, CompressedBufferImage&, BufferUsage)
          * for more information.
          * @requires_gl45 Extension @extension{ARB,get_texture_sub_image}
+         * @requires_gl43 Extension @extension{ARB,internalformat_query2} if
+         *      @ref CompressedPixelStorage::compressedBlockSize() and
+         *      @ref CompressedPixelStorage::compressedBlockDataSize() are not
+         *      set to non-zero values
          * @requires_gl Texture image queries are not available in OpenGL ES or
          *      WebGL. See @ref Framebuffer::read() for possible workaround.
          */
diff --git a/src/Magnum/CubeMapTextureArray.h b/src/Magnum/CubeMapTextureArray.h
index f0f159ebf..c72f76979 100644
--- a/src/Magnum/CubeMapTextureArray.h
+++ b/src/Magnum/CubeMapTextureArray.h
@@ -583,6 +583,10 @@ class MAGNUM_EXPORT CubeMapTextureArray: public AbstractTexture {
          * See @ref Texture::compressedSubImage(Int, const RangeTypeFor<dimensions, Int>&, CompressedImage&)
          * for more information.
          * @requires_gl45 Extension @extension{ARB,get_texture_sub_image}
+         * @requires_gl43 Extension @extension{ARB,internalformat_query2} if
+         *      @ref CompressedPixelStorage::compressedBlockSize() and
+         *      @ref CompressedPixelStorage::compressedBlockDataSize() are not
+         *      set to non-zero values
          * @requires_gl Texture image queries are not available in OpenGL ES.
          *      See @ref Framebuffer::read() for possible workaround.
          */
@@ -605,6 +609,10 @@ class MAGNUM_EXPORT CubeMapTextureArray: public AbstractTexture {
          * See @ref Texture::compressedSubImage(Int, const RangeTypeFor<dimensions, Int>&, CompressedBufferImage&, BufferUsage)
          * for more information.
          * @requires_gl45 Extension @extension{ARB,get_texture_sub_image}
+         * @requires_gl43 Extension @extension{ARB,internalformat_query2} if
+         *      @ref CompressedPixelStorage::compressedBlockSize() and
+         *      @ref CompressedPixelStorage::compressedBlockDataSize() are not
+         *      set to non-zero values
          * @requires_gl Texture image queries are not available in OpenGL ES.
          *      See @ref Framebuffer::read() for possible workaround.
          */
diff --git a/src/Magnum/PixelStorage.h b/src/Magnum/PixelStorage.h
index 083ea2846..84ae2ee7e 100644
--- a/src/Magnum/PixelStorage.h
+++ b/src/Magnum/PixelStorage.h
@@ -444,9 +444,8 @@ namespace Implementation {
     #endif
 
     #ifndef MAGNUM_TARGET_GLES
-    template<std::size_t dimensions, class T> std::pair<std::size_t, std::size_t> compressedImageDataOffsetSizeFor(const T& image, const Math::Vector<dimensions, Int>& size, std::size_t dataSize) {
-        if(!image.storage().compressedBlockSize().product() || !image.storage().compressedBlockDataSize())
-            return {0, dataSize};
+    template<std::size_t dimensions, class T> std::pair<std::size_t, std::size_t> compressedImageDataOffsetSizeFor(const T& image, const Math::Vector<dimensions, Int>& size) {
+        CORRADE_INTERNAL_ASSERT(image.storage().compressedBlockSize().product() && image.storage().compressedBlockDataSize());
 
         Math::Vector3<std::size_t> offset, blockCount;
         std::size_t blockDataSize;
@@ -458,14 +457,15 @@ namespace Implementation {
     }
 
     /* Used in image query functions */
-    template<std::size_t dimensions, class T> std::size_t compressedImageDataSizeFor(const T& image, const Math::Vector<dimensions, Int>& size, std::size_t dataSize) {
-        auto r = compressedImageDataOffsetSizeFor(image, size, dataSize);
+    template<std::size_t dimensions, class T> std::size_t compressedImageDataSizeFor(const T& image, const Math::Vector<dimensions, Int>& size) {
+        auto r = compressedImageDataOffsetSizeFor(image, size);
         return r.first + r.second;
     }
 
     /* Use in compressed image upload functions */
     template<class T> std::size_t occupiedCompressedImageDataSize(const T& image, std::size_t dataSize) {
-        return compressedImageDataOffsetSizeFor(image, image.size(), dataSize).second;
+        return image.storage().compressedBlockSize().product() && image.storage().compressedBlockDataSize()
+            ? compressedImageDataOffsetSizeFor(image, image.size()).second : dataSize;
     }
     #else
     template<class T> std::size_t occupiedCompressedImageDataSize(const T&, std::size_t dataSize) {
diff --git a/src/Magnum/RectangleTexture.h b/src/Magnum/RectangleTexture.h
index f1698d09b..0475f3650 100644
--- a/src/Magnum/RectangleTexture.h
+++ b/src/Magnum/RectangleTexture.h
@@ -442,6 +442,10 @@ class MAGNUM_EXPORT RectangleTexture: public AbstractTexture {
          * See @ref Texture::compressedSubImage(Int, const RangeTypeFor<dimensions, Int>&, CompressedImage&)
          * for more information.
          * @requires_gl45 Extension @extension{ARB,get_texture_sub_image}
+         * @requires_gl43 Extension @extension{ARB,internalformat_query2} if
+         *      @ref CompressedPixelStorage::compressedBlockSize() and
+         *      @ref CompressedPixelStorage::compressedBlockDataSize() are not
+         *      set to non-zero values
          */
         void compressedSubImage(const Range2Di& range, CompressedImage2D& image) {
             AbstractTexture::compressedSubImage<2>(0, range, image);
@@ -462,6 +466,10 @@ class MAGNUM_EXPORT RectangleTexture: public AbstractTexture {
          * See @ref Texture::compressedSubImage(Int, const RangeTypeFor<dimensions, Int>&, CompressedBufferImage&, BufferUsage)
          * for more information.
          * @requires_gl45 Extension @extension{ARB,get_texture_sub_image}
+         * @requires_gl43 Extension @extension{ARB,internalformat_query2} if
+         *      @ref CompressedPixelStorage::compressedBlockSize() and
+         *      @ref CompressedPixelStorage::compressedBlockDataSize() are not
+         *      set to non-zero values
          */
         void compressedSubImage(const Range2Di& range, CompressedBufferImage2D& image, BufferUsage usage) {
             AbstractTexture::compressedSubImage<2>(0, range, image, usage);
diff --git a/src/Magnum/Test/CubeMapTextureArrayGLTest.cpp b/src/Magnum/Test/CubeMapTextureArrayGLTest.cpp
index f808dacc8..785851c17 100644
--- a/src/Magnum/Test/CubeMapTextureArrayGLTest.cpp
+++ b/src/Magnum/Test/CubeMapTextureArrayGLTest.cpp
@@ -1032,14 +1032,14 @@ void CubeMapTextureArrayGLTest::compressedSubImageQuery() {
 
     if(!Context::current().isExtensionSupported<Extensions::GL::ARB::texture_cube_map_array>())
         CORRADE_SKIP(Extensions::GL::ARB::texture_cube_map_array::string() + std::string(" is not supported."));
-    if(!Context::current().isExtensionSupported<Extensions::GL::ARB::internalformat_query2>())
-        CORRADE_SKIP(Extensions::GL::ARB::internalformat_query2::string() + std::string(" is not supported."));
     if(!Context::current().isExtensionSupported<Extensions::GL::ARB::get_texture_sub_image>())
         CORRADE_SKIP(Extensions::GL::ARB::get_texture_sub_image::string() + std::string(" is not supported."));
     if(!Context::current().isExtensionSupported<Extensions::GL::EXT::texture_compression_s3tc>())
         CORRADE_SKIP(Extensions::GL::EXT::texture_compression_s3tc::string() + std::string(" is not supported."));
-    if(CompressedPixelStorageData[testCaseInstanceId()].storage != CompressedPixelStorage{} && !Context::current().isExtensionSupported<Extensions::GL::ARB::compressed_texture_pixel_storage>())
+    if(CompressedSubPixelStorageData[testCaseInstanceId()].storage != CompressedPixelStorage{} && !Context::current().isExtensionSupported<Extensions::GL::ARB::compressed_texture_pixel_storage>())
         CORRADE_SKIP(Extensions::GL::ARB::compressed_texture_pixel_storage::string() + std::string(" is not supported."));
+    if(CompressedSubPixelStorageData[testCaseInstanceId()].storage == CompressedPixelStorage{} && !Context::current().isExtensionSupported<Extensions::GL::ARB::internalformat_query2>())
+        CORRADE_SKIP(Extensions::GL::ARB::internalformat_query2::string() + std::string(" is not supported."));
 
     CubeMapTextureArray texture;
     texture.setStorage(1, TextureFormat::CompressedRGBAS3tcDxt3, {12, 12, 6})
@@ -1062,14 +1062,14 @@ void CubeMapTextureArrayGLTest::compressedSubImageQueryBuffer() {
 
     if(!Context::current().isExtensionSupported<Extensions::GL::ARB::texture_cube_map_array>())
         CORRADE_SKIP(Extensions::GL::ARB::texture_cube_map_array::string() + std::string(" is not supported."));
-    if(!Context::current().isExtensionSupported<Extensions::GL::ARB::internalformat_query2>())
-        CORRADE_SKIP(Extensions::GL::ARB::internalformat_query2::string() + std::string(" is not supported."));
     if(!Context::current().isExtensionSupported<Extensions::GL::ARB::get_texture_sub_image>())
         CORRADE_SKIP(Extensions::GL::ARB::get_texture_sub_image::string() + std::string(" is not supported."));
     if(!Context::current().isExtensionSupported<Extensions::GL::EXT::texture_compression_s3tc>())
         CORRADE_SKIP(Extensions::GL::EXT::texture_compression_s3tc::string() + std::string(" is not supported."));
-    if(CompressedPixelStorageData[testCaseInstanceId()].storage != CompressedPixelStorage{} && !Context::current().isExtensionSupported<Extensions::GL::ARB::compressed_texture_pixel_storage>())
+    if(CompressedSubPixelStorageData[testCaseInstanceId()].storage != CompressedPixelStorage{} && !Context::current().isExtensionSupported<Extensions::GL::ARB::compressed_texture_pixel_storage>())
         CORRADE_SKIP(Extensions::GL::ARB::compressed_texture_pixel_storage::string() + std::string(" is not supported."));
+    if(CompressedSubPixelStorageData[testCaseInstanceId()].storage == CompressedPixelStorage{} && !Context::current().isExtensionSupported<Extensions::GL::ARB::internalformat_query2>())
+        CORRADE_SKIP(Extensions::GL::ARB::internalformat_query2::string() + std::string(" is not supported."));
 
     CubeMapTextureArray texture;
     texture.setStorage(1, TextureFormat::CompressedRGBAS3tcDxt3, {12, 12, 6})
diff --git a/src/Magnum/Test/CubeMapTextureGLTest.cpp b/src/Magnum/Test/CubeMapTextureGLTest.cpp
index ca6af1762..74295baed 100644
--- a/src/Magnum/Test/CubeMapTextureGLTest.cpp
+++ b/src/Magnum/Test/CubeMapTextureGLTest.cpp
@@ -1103,6 +1103,8 @@ void CubeMapTextureGLTest::compressedSubImageQuery() {
         CORRADE_SKIP(Extensions::GL::EXT::texture_compression_s3tc::string() + std::string(" is not supported."));
     if(CompressedPixelStorageData[testCaseInstanceId()].storage != CompressedPixelStorage{} && !Context::current().isExtensionSupported<Extensions::GL::ARB::compressed_texture_pixel_storage>())
         CORRADE_SKIP(Extensions::GL::ARB::compressed_texture_pixel_storage::string() + std::string(" is not supported."));
+    if(CompressedPixelStorageData[testCaseInstanceId()].storage == CompressedPixelStorage{} && !Context::current().isExtensionSupported<Extensions::GL::ARB::internalformat_query2>())
+        CORRADE_SKIP(Extensions::GL::ARB::internalformat_query2::string() + std::string(" is not supported."));
 
     CubeMapTexture texture;
     texture.setStorage(1, TextureFormat::CompressedRGBAS3tcDxt3, Vector2i{12})
@@ -1132,6 +1134,8 @@ void CubeMapTextureGLTest::compressedSubImageQueryBuffer() {
         CORRADE_SKIP(Extensions::GL::EXT::texture_compression_s3tc::string() + std::string(" is not supported."));
     if(CompressedPixelStorageData[testCaseInstanceId()].storage != CompressedPixelStorage{} && !Context::current().isExtensionSupported<Extensions::GL::ARB::compressed_texture_pixel_storage>())
         CORRADE_SKIP(Extensions::GL::ARB::compressed_texture_pixel_storage::string() + std::string(" is not supported."));
+    if(CompressedPixelStorageData[testCaseInstanceId()].storage == CompressedPixelStorage{} && !Context::current().isExtensionSupported<Extensions::GL::ARB::internalformat_query2>())
+        CORRADE_SKIP(Extensions::GL::ARB::internalformat_query2::string() + std::string(" is not supported."));
 
     CubeMapTexture texture;
     texture.setStorage(1, TextureFormat::CompressedRGBAS3tcDxt3, Vector2i{12})
diff --git a/src/Magnum/Test/PixelStorageTest.cpp b/src/Magnum/Test/PixelStorageTest.cpp
index e21fbef0a..93b9b1101 100644
--- a/src/Magnum/Test/PixelStorageTest.cpp
+++ b/src/Magnum/Test/PixelStorageTest.cpp
@@ -345,11 +345,6 @@ void PixelStorageTest::dataPropertiesCompressedImageHeight() {
 }
 
 void PixelStorageTest::dataOffsetSizeCompressed() {
-    /* Tf the storage doesn't contain any info about block sizes (the default),
-       using the provided value */
-    CORRADE_COMPARE(Implementation::compressedImageDataOffsetSizeFor(CompressedImage3D{},
-        Vector2i{37, 35}, 1579), (std::pair<std::size_t, std::size_t>{0, 1579}));
-
     /* The same parameters as in PixelStorageGLTest 3D case */
     const CompressedImage3D image{CompressedPixelStorage{}
         .setCompressedBlockSize({4, 4, 1})
@@ -357,7 +352,7 @@ void PixelStorageTest::dataOffsetSizeCompressed() {
         .setRowLength(8)
         .setImageHeight(8)
         .setSkip({4, 4, 4})};
-    CORRADE_COMPARE(Implementation::compressedImageDataOffsetSizeFor(image, Vector3i{4, 4, 1}, 1579),
+    CORRADE_COMPARE(Implementation::compressedImageDataOffsetSizeFor(image, Vector3i{4, 4, 1}),
         (std::pair<std::size_t, std::size_t>{16*4*4 + 16*2 + 16, 16}));
 }
 #endif
diff --git a/src/Magnum/Test/TextureArrayGLTest.cpp b/src/Magnum/Test/TextureArrayGLTest.cpp
index 812d8f29a..c2d4c08c1 100644
--- a/src/Magnum/Test/TextureArrayGLTest.cpp
+++ b/src/Magnum/Test/TextureArrayGLTest.cpp
@@ -1434,6 +1434,8 @@ void TextureArrayGLTest::compressedSubImage2DQuery() {
         CORRADE_SKIP(Extensions::GL::EXT::texture_compression_s3tc::string() + std::string(" is not supported."));
     if(CompressedPixelStorage2DData[testCaseInstanceId()].storage != CompressedPixelStorage{} && !Context::current().isExtensionSupported<Extensions::GL::ARB::compressed_texture_pixel_storage>())
         CORRADE_SKIP(Extensions::GL::ARB::compressed_texture_pixel_storage::string() + std::string(" is not supported."));
+    if(CompressedPixelStorage2DData[testCaseInstanceId()].storage == CompressedPixelStorage{} && !Context::current().isExtensionSupported<Extensions::GL::ARB::internalformat_query2>())
+        CORRADE_SKIP(Extensions::GL::ARB::internalformat_query2::string() + std::string(" is not supported."));
 
     Texture2DArray texture;
     texture.setStorage(1, TextureFormat::CompressedRGBAS3tcDxt3, {12, 4, 4})
@@ -1463,6 +1465,8 @@ void TextureArrayGLTest::compressedSubImage2DQueryBuffer() {
         CORRADE_SKIP(Extensions::GL::EXT::texture_compression_s3tc::string() + std::string(" is not supported."));
     if(CompressedPixelStorage2DData[testCaseInstanceId()].storage != CompressedPixelStorage{} && !Context::current().isExtensionSupported<Extensions::GL::ARB::compressed_texture_pixel_storage>())
         CORRADE_SKIP(Extensions::GL::ARB::compressed_texture_pixel_storage::string() + std::string(" is not supported."));
+    if(CompressedPixelStorage2DData[testCaseInstanceId()].storage == CompressedPixelStorage{} && !Context::current().isExtensionSupported<Extensions::GL::ARB::internalformat_query2>())
+        CORRADE_SKIP(Extensions::GL::ARB::internalformat_query2::string() + std::string(" is not supported."));
 
     Texture2DArray texture;
     texture.setStorage(1, TextureFormat::CompressedRGBAS3tcDxt3, {12, 4, 4})
diff --git a/src/Magnum/Test/TextureGLTest.cpp b/src/Magnum/Test/TextureGLTest.cpp
index 184e3b3ed..0889d8a74 100644
--- a/src/Magnum/Test/TextureGLTest.cpp
+++ b/src/Magnum/Test/TextureGLTest.cpp
@@ -1753,6 +1753,8 @@ void TextureGLTest::compressedSubImage2DQuery() {
         CORRADE_SKIP(Extensions::GL::EXT::texture_compression_s3tc::string() + std::string(" is not supported."));
     if(CompressedPixelStorage2DData[testCaseInstanceId()].storage != CompressedPixelStorage{} && !Context::current().isExtensionSupported<Extensions::GL::ARB::compressed_texture_pixel_storage>())
         CORRADE_SKIP(Extensions::GL::ARB::compressed_texture_pixel_storage::string() + std::string(" is not supported."));
+    if(CompressedPixelStorage2DData[testCaseInstanceId()].storage == CompressedPixelStorage{} && !Context::current().isExtensionSupported<Extensions::GL::ARB::internalformat_query2>())
+        CORRADE_SKIP(Extensions::GL::ARB::internalformat_query2::string() + std::string(" is not supported."));
 
     Texture2D texture;
     texture.setStorage(1, TextureFormat::CompressedRGBAS3tcDxt3, {12, 4})
@@ -1780,6 +1782,8 @@ void TextureGLTest::compressedSubImage2DQueryBuffer() {
         CORRADE_SKIP(Extensions::GL::EXT::texture_compression_s3tc::string() + std::string(" is not supported."));
     if(CompressedPixelStorage2DData[testCaseInstanceId()].storage != CompressedPixelStorage{} && !Context::current().isExtensionSupported<Extensions::GL::ARB::compressed_texture_pixel_storage>())
         CORRADE_SKIP(Extensions::GL::ARB::compressed_texture_pixel_storage::string() + std::string(" is not supported."));
+    if(CompressedPixelStorage2DData[testCaseInstanceId()].storage == CompressedPixelStorage{} && !Context::current().isExtensionSupported<Extensions::GL::ARB::internalformat_query2>())
+        CORRADE_SKIP(Extensions::GL::ARB::internalformat_query2::string() + std::string(" is not supported."));
 
     Texture2D texture;
     texture.setStorage(1, TextureFormat::CompressedRGBAS3tcDxt3, {12, 4})
@@ -2199,6 +2203,8 @@ void TextureGLTest::compressedSubImage3DQuery() {
         CORRADE_SKIP(Extensions::GL::ARB::texture_compression_bptc::string() + std::string(" is not supported."));
     if(CompressedPixelStorage3DData[testCaseInstanceId()].storage != CompressedPixelStorage{} && !Context::current().isExtensionSupported<Extensions::GL::ARB::compressed_texture_pixel_storage>())
         CORRADE_SKIP(Extensions::GL::ARB::compressed_texture_pixel_storage::string() + std::string(" is not supported."));
+    if(CompressedPixelStorage3DData[testCaseInstanceId()].storage == CompressedPixelStorage{} && !Context::current().isExtensionSupported<Extensions::GL::ARB::internalformat_query2>())
+        CORRADE_SKIP(Extensions::GL::ARB::internalformat_query2::string() + std::string(" is not supported."));
 
     Texture3D texture;
     texture.setStorage(1, TextureFormat::CompressedRGBABptcUnorm, {12, 4, 4})
@@ -2232,6 +2238,8 @@ void TextureGLTest::compressedSubImage3DQueryBuffer() {
         CORRADE_SKIP(Extensions::GL::ARB::texture_compression_bptc::string() + std::string(" is not supported."));
     if(CompressedPixelStorage3DData[testCaseInstanceId()].storage != CompressedPixelStorage{} && !Context::current().isExtensionSupported<Extensions::GL::ARB::compressed_texture_pixel_storage>())
         CORRADE_SKIP(Extensions::GL::ARB::compressed_texture_pixel_storage::string() + std::string(" is not supported."));
+    if(CompressedPixelStorage3DData[testCaseInstanceId()].storage == CompressedPixelStorage{} && !Context::current().isExtensionSupported<Extensions::GL::ARB::internalformat_query2>())
+        CORRADE_SKIP(Extensions::GL::ARB::internalformat_query2::string() + std::string(" is not supported."));
 
     Texture3D texture;
     texture.setStorage(1, TextureFormat::CompressedRGBABptcUnorm, {12, 4, 4})
diff --git a/src/Magnum/Texture.h b/src/Magnum/Texture.h
index c1aed93de..71e0040b6 100644
--- a/src/Magnum/Texture.h
+++ b/src/Magnum/Texture.h
@@ -129,7 +129,7 @@ template<UnsignedInt dimensions> class Texture: public AbstractTexture {
          *
          * If @p format is compressed, returns compressed block size (in
          * pixels). For uncompressed formats returns zero vector.
-         * @see @ref compressedBlockDataSize(), @fn_gl{Getinternalformat} with
+         * @see @ref compressedBlockDataSize(), @fn_gl{GetInternalformat} with
          *      @def_gl{TEXTURE_COMPRESSED_BLOCK_WIDTH},
          *      @def_gl{TEXTURE_COMPRESSED_BLOCK_HEIGHT}
          * @requires_gl43 Extension @extension{ARB,internalformat_query2}
@@ -145,7 +145,7 @@ template<UnsignedInt dimensions> class Texture: public AbstractTexture {
          *
          * If @p format is compressed, returns compressed block data size (in
          * bytes). For uncompressed formats returns zero.
-         * @see @ref compressedBlockSize(), @fn_gl{Getinternalformat} with
+         * @see @ref compressedBlockSize(), @fn_gl{GetInternalformat} with
          *      @def_gl{TEXTURE_COMPRESSED_BLOCK_SIZE}
          * @requires_gl43 Extension @extension{ARB,internalformat_query2}
          * @requires_gl Compressed texture queries are not available in OpenGL
@@ -958,9 +958,16 @@ template<UnsignedInt dimensions> class Texture: public AbstractTexture {
          * @see @fn_gl2{GetTextureLevelParameter,GetTexLevelParameter},
          *      @fn_gl_extension{GetTextureLevelParameter,EXT,direct_state_access},
          *      eventually @fn_gl{GetTexLevelParameter} with
-         *      @def_gl{TEXTURE_COMPRESSED_IMAGE_SIZE}, @def_gl{TEXTURE_INTERNAL_FORMAT},
-         *      then @fn_gl{GetCompressedTextureSubImage}
+         *      @def_gl{TEXTURE_INTERNAL_FORMAT}, then possibly
+         *      @fn_gl{GetInternalformat} with @def_gl{TEXTURE_COMPRESSED_BLOCK_SIZE},
+         *      @def_gl{TEXTURE_COMPRESSED_BLOCK_WIDTH} and
+         *      @def_gl{TEXTURE_COMPRESSED_BLOCK_HEIGHT}, then
+         *      @fn_gl{GetCompressedTextureSubImage}
          * @requires_gl45 Extension @extension{ARB,get_texture_sub_image}
+         * @requires_gl43 Extension @extension{ARB,internalformat_query2} if
+         *      @ref CompressedPixelStorage::compressedBlockSize() and
+         *      @ref CompressedPixelStorage::compressedBlockDataSize() are not
+         *      set to non-zero values
          * @requires_gl Texture image queries are not available in OpenGL ES or
          *      WebGL. See @ref Framebuffer::read() for possible workaround.
          */
@@ -987,6 +994,10 @@ template<UnsignedInt dimensions> class Texture: public AbstractTexture {
          * See @ref compressedSubImage(Int, const RangeTypeFor<dimensions, Int>&, CompressedBufferImage&, BufferUsage)
          * for more information.
          * @requires_gl45 Extension @extension{ARB,get_texture_sub_image}
+         * @requires_gl43 Extension @extension{ARB,internalformat_query2} if
+         *      @ref CompressedPixelStorage::compressedBlockSize() and
+         *      @ref CompressedPixelStorage::compressedBlockDataSize() are not
+         *      set to non-zero values
          * @requires_gl Texture image queries are not available in OpenGL ES or
          *      WebGL. See @ref Framebuffer::read() for possible workaround.
          */
diff --git a/src/Magnum/TextureArray.h b/src/Magnum/TextureArray.h
index 1fb77181d..526c491e8 100644
--- a/src/Magnum/TextureArray.h
+++ b/src/Magnum/TextureArray.h
@@ -606,6 +606,10 @@ template<UnsignedInt dimensions> class TextureArray: public AbstractTexture {
          * See @ref Texture::compressedSubImage(Int, const RangeTypeFor<dimensions, Int>&, CompressedImage&)
          * for more information.
          * @requires_gl45 Extension @extension{ARB,get_texture_sub_image}
+         * @requires_gl43 Extension @extension{ARB,internalformat_query2} if
+         *      @ref CompressedPixelStorage::compressedBlockSize() and
+         *      @ref CompressedPixelStorage::compressedBlockDataSize() are not
+         *      set to non-zero values
          * @requires_gl Texture image queries are not available in OpenGL ES or
          *      WebGL. See @ref Framebuffer::read() for possible workaround.
          */
@@ -628,6 +632,10 @@ template<UnsignedInt dimensions> class TextureArray: public AbstractTexture {
          * See @ref Texture::compressedSubImage(Int, const RangeTypeFor<dimensions, Int>&, CompressedBufferImage&, BufferUsage)
          * for more information.
          * @requires_gl45 Extension @extension{ARB,get_texture_sub_image}
+         * @requires_gl43 Extension @extension{ARB,internalformat_query2} if
+         *      @ref CompressedPixelStorage::compressedBlockSize() and
+         *      @ref CompressedPixelStorage::compressedBlockDataSize() are not
+         *      set to non-zero values
          * @requires_gl Texture image queries are not available in OpenGL ES or
          *      WebGL. See @ref Framebuffer::read() for possible workaround.
          */