diff --git a/Lib/test/support/__init__.py b/Lib/test/support/__init__.py
--- a/Lib/test/support/__init__.py
+++ b/Lib/test/support/__init__.py
@@ -1295,16 +1295,17 @@ def transient_internet(resource_name, *,
     """Return a context manager that raises ResourceDenied when various issues
     with the Internet connection manifest themselves as exceptions."""
     default_errnos = [
         ('ECONNREFUSED', 111),
         ('ECONNRESET', 104),
         ('EHOSTUNREACH', 113),
         ('ENETUNREACH', 101),
         ('ETIMEDOUT', 110),
+        ('ENOTCONN', 107),
     ]
     default_gai_errnos = [
         ('EAI_AGAIN', -3),
         ('EAI_FAIL', -4),
         ('EAI_NONAME', -2),
         ('EAI_NODATA', -5),
         # Encountered when trying to resolve IPv6-only hostnames
         ('WSANO_DATA', 11004),
diff --git a/Lib/test/test_ssl.py b/Lib/test/test_ssl.py
--- a/Lib/test/test_ssl.py
+++ b/Lib/test/test_ssl.py
@@ -1373,17 +1373,21 @@ class NetworkedTests(unittest.TestCase):
 
     def test_connect_ex(self):
         # Issue #11326: check connect_ex() implementation
         with support.transient_internet("svn.python.org"):
             s = ssl.wrap_socket(socket.socket(socket.AF_INET),
                                 cert_reqs=ssl.CERT_REQUIRED,
                                 ca_certs=SVN_PYTHON_ORG_ROOT_CERT)
             try:
-                self.assertEqual(0, s.connect_ex(("svn.python.org", 443)))
+                rc = s.connect_ex(("svn.python.org", 443))
+                if rc == errno.ECONNREFUSED:
+                    self.skipTest("svn.python.org cannot be reached: "
+                                  "Connection refused")
+                self.assertEqual(rc, 0)
                 self.assertTrue(s.getpeercert())
             finally:
                 s.close()
 
     def test_non_blocking_connect_ex(self):
         # Issue #11326: non-blocking connect_ex() should allow handshake
         # to proceed after the socket gets ready.
         with support.transient_internet("svn.python.org"):
@@ -1420,16 +1424,19 @@ class NetworkedTests(unittest.TestCase):
                                 cert_reqs=ssl.CERT_REQUIRED,
                                 ca_certs=SVN_PYTHON_ORG_ROOT_CERT,
                                 do_handshake_on_connect=False)
             try:
                 s.settimeout(0.0000001)
                 rc = s.connect_ex(('svn.python.org', 443))
                 if rc == 0:
                     self.skipTest("svn.python.org responded too quickly")
+                if rc == errno.ECONNREFUSED:
+                    self.skipTest("svn.python.org cannot be reached: "
+                                  "Connection refused")
                 self.assertIn(rc, (errno.EAGAIN, errno.EWOULDBLOCK))
             finally:
                 s.close()
 
     def test_connect_ex_error(self):
         with support.transient_internet("svn.python.org"):
             s = ssl.wrap_socket(socket.socket(socket.AF_INET),
                                 cert_reqs=ssl.CERT_REQUIRED,