diff -r 955547e57cff Lib/tarfile.py
--- a/Lib/tarfile.py	Sun Mar 13 19:32:21 2011 +0100
+++ b/Lib/tarfile.py	Tue Mar 15 14:09:30 2011 -0400
@@ -1804,11 +1804,11 @@
             fileobj = gzip.GzipFile(name, mode + "b", compresslevel, fileobj)
             t = cls.taropen(name, mode, fileobj, **kwargs)
         except IOError:
-            if not extfileobj:
+            if not extfileobj and fileobj is not None:
                 fileobj.close()
             raise ReadError("not a gzip file")
         except:
-            if not extfileobj:
+            if not extfileobj and fileobj is not None:
                 fileobj.close()
             raise
         t._extfileobj = extfileobj
diff -r 955547e57cff Lib/test/test_tarfile.py
--- a/Lib/test/test_tarfile.py	Sun Mar 13 19:32:21 2011 +0100
+++ b/Lib/test/test_tarfile.py	Tue Mar 15 14:09:30 2011 -0400
@@ -1652,6 +1652,12 @@
 class GzipMiscReadTest(MiscReadTest):
     tarname = gzipname
     mode = "r:gz"
+
+    def test_non_existent_targz_file(self):
+        # Test for issue11513: prevent non-existent gzipped tarfiles raising
+        # multiple exceptions.
+        self.assertRaises(tarfile.ReadError, tarfile.open, "xxx", self.mode)
+
 class GzipUstarReadTest(UstarReadTest):
     tarname = gzipname
     mode = "r:gz"