diff --git a/src/Magnum/GL/Test/CubeMapTextureGLTest.cpp b/src/Magnum/GL/Test/CubeMapTextureGLTest.cpp
index 31ca434d9..c9340e36e 100644
--- a/src/Magnum/GL/Test/CubeMapTextureGLTest.cpp
+++ b/src/Magnum/GL/Test/CubeMapTextureGLTest.cpp
@@ -1265,9 +1265,13 @@ void CubeMapTextureGLTest::fullImageQuery() {
     MAGNUM_VERIFY_NO_GL_ERROR();
 
     CORRADE_COMPARE(image.size(), Vector3i(2, 2, 6));
-    CORRADE_COMPARE_AS(Containers::arrayCast<UnsignedByte>(image.data()).suffix(FullPixelStorageData[testCaseInstanceId()].offset),
-        FullPixelStorageData[testCaseInstanceId()].data,
-        TestSuite::Compare::Container);
+    {
+        CORRADE_EXPECT_FAIL_IF((Context::current().detectedDriver() & Context::DetectedDriver::Mesa) && FullPixelStorageData[testCaseInstanceId()].storage != PixelStorage{},
+            "Mesa drivers can't handle non-default pixel storage for full cubemap image queries.");
+        CORRADE_COMPARE_AS(Containers::arrayCast<UnsignedByte>(image.data()).suffix(FullPixelStorageData[testCaseInstanceId()].offset),
+            FullPixelStorageData[testCaseInstanceId()].data,
+            TestSuite::Compare::Container);
+    }
 }
 
 void CubeMapTextureGLTest::fullImageQueryBuffer() {
@@ -1293,8 +1297,12 @@ void CubeMapTextureGLTest::fullImageQueryBuffer() {
 
     CORRADE_COMPARE(image.size(), Vector3i(2, 2, 6));
     const auto imageData = image.buffer().data();
-    CORRADE_COMPARE_AS(Containers::arrayCast<UnsignedByte>(imageData).suffix(FullPixelStorageData[testCaseInstanceId()].offset),
-        FullPixelStorageData[testCaseInstanceId()].data, TestSuite::Compare::Container);
+    {
+        CORRADE_EXPECT_FAIL_IF((Context::current().detectedDriver() & Context::DetectedDriver::Mesa) && FullPixelStorageData[testCaseInstanceId()].storage != PixelStorage{},
+            "Mesa drivers can't handle non-default pixel storage for full cubemap image queries.");
+        CORRADE_COMPARE_AS(Containers::arrayCast<UnsignedByte>(imageData).suffix(FullPixelStorageData[testCaseInstanceId()].offset),
+            FullPixelStorageData[testCaseInstanceId()].data, TestSuite::Compare::Container);
+    }
 }
 
 void CubeMapTextureGLTest::compressedFullImageQuery() {
@@ -1320,9 +1328,13 @@ void CubeMapTextureGLTest::compressedFullImageQuery() {
     MAGNUM_VERIFY_NO_GL_ERROR();
 
     CORRADE_COMPARE(image.size(), (Vector3i{4, 4, 6}));
-    CORRADE_COMPARE_AS(Containers::arrayCast<UnsignedByte>(image.data()).suffix(CompressedFullPixelStorageData[testCaseInstanceId()].offset),
-        CompressedFullPixelStorageData[testCaseInstanceId()].data,
-        TestSuite::Compare::Container);
+    {
+        CORRADE_EXPECT_FAIL_IF((Context::current().detectedDriver() & Context::DetectedDriver::Mesa) && CompressedFullPixelStorageData[testCaseInstanceId()].storage != CompressedPixelStorage{},
+            "Mesa drivers can't handle non-default pixel storage for full cubemap image queries.");
+        CORRADE_COMPARE_AS(Containers::arrayCast<UnsignedByte>(image.data()).suffix(CompressedFullPixelStorageData[testCaseInstanceId()].offset),
+            CompressedFullPixelStorageData[testCaseInstanceId()].data,
+            TestSuite::Compare::Container);
+    }
 }
 
 void CubeMapTextureGLTest::compressedFullImageQueryBuffer() {
@@ -1350,9 +1362,13 @@ void CubeMapTextureGLTest::compressedFullImageQueryBuffer() {
 
     CORRADE_COMPARE(image.size(), (Vector3i{4, 4, 6}));
     const auto imageData = image.buffer().data();
-    CORRADE_COMPARE_AS(Containers::arrayCast<UnsignedByte>(imageData).suffix(CompressedFullPixelStorageData[testCaseInstanceId()].offset),
-        CompressedFullPixelStorageData[testCaseInstanceId()].data,
-        TestSuite::Compare::Container);
+    {
+        CORRADE_EXPECT_FAIL_IF((Context::current().detectedDriver() & Context::DetectedDriver::Mesa) && CompressedFullPixelStorageData[testCaseInstanceId()].storage != CompressedPixelStorage{},
+            "Mesa drivers can't handle non-default pixel storage for full cubemap image queries.");
+        CORRADE_COMPARE_AS(Containers::arrayCast<UnsignedByte>(imageData).suffix(CompressedFullPixelStorageData[testCaseInstanceId()].offset),
+            CompressedFullPixelStorageData[testCaseInstanceId()].data,
+            TestSuite::Compare::Container);
+    }
 }
 #endif
 
diff --git a/src/Magnum/GL/Test/TextureGLTest.cpp b/src/Magnum/GL/Test/TextureGLTest.cpp
index 50b9cf92a..388885bc3 100644
--- a/src/Magnum/GL/Test/TextureGLTest.cpp
+++ b/src/Magnum/GL/Test/TextureGLTest.cpp
@@ -2148,9 +2148,13 @@ void TextureGLTest::compressedImage3D() {
     MAGNUM_VERIFY_NO_GL_ERROR();
 
     CORRADE_COMPARE(image.size(), Vector3i{4});
-    CORRADE_COMPARE_AS(Containers::arrayCast<UnsignedByte>(image.data()).suffix(CompressedPixelStorage3DData[testCaseInstanceId()].offset),
-        CompressedPixelStorage3DData[testCaseInstanceId()].data,
-        TestSuite::Compare::Container);
+    {
+        CORRADE_EXPECT_FAIL_IF((Context::current().detectedDriver() & Context::DetectedDriver::Mesa) && CompressedPixelStorage3DData[testCaseInstanceId()].storage != CompressedPixelStorage{},
+            "Mesa drivers can't handle non-default compressed 3D pixel storage.");
+        CORRADE_COMPARE_AS(Containers::arrayCast<UnsignedByte>(image.data()).suffix(CompressedPixelStorage3DData[testCaseInstanceId()].offset),
+            CompressedPixelStorage3DData[testCaseInstanceId()].data,
+            TestSuite::Compare::Container);
+    }
     #endif
 }
 
@@ -2182,9 +2186,13 @@ void TextureGLTest::compressedImage3DBuffer() {
     MAGNUM_VERIFY_NO_GL_ERROR();
 
     CORRADE_COMPARE(image.size(), Vector3i{4});
-    CORRADE_COMPARE_AS(Containers::arrayCast<UnsignedByte>(imageData).suffix(CompressedPixelStorage3DData[testCaseInstanceId()].offset),
-        CompressedPixelStorage3DData[testCaseInstanceId()].data,
-        TestSuite::Compare::Container);
+    {
+        CORRADE_EXPECT_FAIL_IF((Context::current().detectedDriver() & Context::DetectedDriver::Mesa) && CompressedPixelStorage3DData[testCaseInstanceId()].storage != CompressedPixelStorage{},
+            "Mesa drivers can't handle non-default compressed 3D pixel storage.");
+        CORRADE_COMPARE_AS(Containers::arrayCast<UnsignedByte>(imageData).suffix(CompressedPixelStorage3DData[testCaseInstanceId()].offset),
+            CompressedPixelStorage3DData[testCaseInstanceId()].data,
+            TestSuite::Compare::Container);
+    }
     #endif
 }
 #endif
@@ -2265,7 +2273,8 @@ void TextureGLTest::compressedSubImage3D() {
     {
         CORRADE_EXPECT_FAIL_IF(CompressedPixelStorage3DData[testCaseInstanceId()].storage == CompressedPixelStorage{} && (Context::current().detectedDriver() & Context::DetectedDriver::NVidia),
             "Default compressed pixel storage behaves weirdly with BPTC compression on NVidia.");
-
+        CORRADE_EXPECT_FAIL_IF((Context::current().detectedDriver() & Context::DetectedDriver::Mesa),
+            "Mesa drivers can't handle compressed 3D pixel storage for subimages.");
         CORRADE_COMPARE_AS(Containers::arrayCast<UnsignedByte>(image.data()),
             Containers::arrayView(CompressedSubData3DComplete),
             TestSuite::Compare::Container);
@@ -2305,7 +2314,8 @@ void TextureGLTest::compressedSubImage3DBuffer() {
     {
         CORRADE_EXPECT_FAIL_IF(CompressedPixelStorage3DData[testCaseInstanceId()].storage == CompressedPixelStorage{} && (Context::current().detectedDriver() & Context::DetectedDriver::NVidia),
             "Default compressed pixel storage behaves weirdly with BPTC compression on NVidia.");
-
+        CORRADE_EXPECT_FAIL_IF((Context::current().detectedDriver() & Context::DetectedDriver::Mesa),
+            "Mesa drivers can't handle compressed 3D pixel storage for subimages.");
         CORRADE_COMPARE_AS(Containers::arrayCast<UnsignedByte>(imageData),
             Containers::arrayView(CompressedSubData3DComplete),
             TestSuite::Compare::Container);
@@ -2343,7 +2353,8 @@ void TextureGLTest::compressedSubImage3DQuery() {
     {
         CORRADE_EXPECT_FAIL_IF(CompressedPixelStorage3DData[testCaseInstanceId()].storage == CompressedPixelStorage{} && (Context::current().detectedDriver() & Context::DetectedDriver::NVidia),
             "Default compressed pixel storage behaves weirdly with BPTC compression on NVidia.");
-
+        CORRADE_EXPECT_FAIL_IF((Context::current().detectedDriver() & Context::DetectedDriver::Mesa),
+            "Mesa drivers can't handle compressed 3D pixel storage for subimages.");
         CORRADE_COMPARE_AS(Containers::arrayCast<UnsignedByte>(image.data()).suffix(CompressedPixelStorage3DData[testCaseInstanceId()].offset),
             CompressedPixelStorage3DData[testCaseInstanceId()].data,
             TestSuite::Compare::Container);
@@ -2379,7 +2390,8 @@ void TextureGLTest::compressedSubImage3DQueryBuffer() {
     {
         CORRADE_EXPECT_FAIL_IF(CompressedPixelStorage3DData[testCaseInstanceId()].storage == CompressedPixelStorage{} && (Context::current().detectedDriver() & Context::DetectedDriver::NVidia),
             "Default compressed pixel storage behaves weirdly with BPTC compression on NVidia.");
-
+        CORRADE_EXPECT_FAIL_IF((Context::current().detectedDriver() & Context::DetectedDriver::Mesa),
+            "Mesa drivers can't handle compressed 3D pixel storage for subimages.");
         CORRADE_COMPARE_AS(Containers::arrayCast<UnsignedByte>(imageData).suffix(CompressedPixelStorage3DData[testCaseInstanceId()].offset),
             CompressedPixelStorage3DData[testCaseInstanceId()].data,
             TestSuite::Compare::Container);