diff --git a/src/tests/gl_tests/MultithreadingTest.cpp b/src/tests/gl_tests/MultithreadingTest.cpp
index d4326703f..c649741bb 100644
--- a/src/tests/gl_tests/MultithreadingTest.cpp
+++ b/src/tests/gl_tests/MultithreadingTest.cpp
@@ -725,7 +725,7 @@ void MultithreadingTestES3::mainThreadDraw(bool useDraw)
 TEST_P(MultithreadingTestES3, MultithreadFenceDraw)
 {
     // http://anglebug.com/5418
-    ANGLE_SKIP_TEST_IF(IsLinux() && IsIntel() && IsVulkan());
+    ANGLE_SKIP_TEST_IF(IsLinux() && IsVulkan() && (IsIntel() || IsSwiftshaderDevice()));
 
     // Have the secondary thread use glDrawArrays()
     mainThreadDraw(true);
diff --git a/src/tests/gl_tests/VulkanMultithreadingTest.cpp b/src/tests/gl_tests/VulkanMultithreadingTest.cpp
index 812778a88..c0a74a176 100644
--- a/src/tests/gl_tests/VulkanMultithreadingTest.cpp
+++ b/src/tests/gl_tests/VulkanMultithreadingTest.cpp
@@ -126,6 +126,8 @@ class VulkanMultithreadingTest : public ANGLETest
 // descriptor pools.
 TEST_P(VulkanMultithreadingTest, MultiContextDrawSmallDescriptorPools)
 {
+    // TODO(http://anglebug.com/6633: Flaky on linux.
+    ANGLE_SKIP_TEST_IF(IsLinux());
     ANGLE_SKIP_TEST_IF(!platformSupportsMultithreading());
     // Verify the extension is enabled.
     ASSERT_TRUE(IsGLExtensionEnabled(kExtensionName));