diff --git a/trio/tests/test_socket.py b/trio/tests/test_socket.py
index c405f7eae7..2150a553ce 100644
--- a/trio/tests/test_socket.py
+++ b/trio/tests/test_socket.py
@@ -4,6 +4,8 @@
 import socket as stdlib_socket
 import inspect
 
+from .._core.tests.tutil import slow
+
 from .. import _core
 from .. import socket as tsocket
 from ..testing import assert_yields, wait_all_tasks_blocked
@@ -549,6 +551,8 @@ def connect(self, *args, **kwargs):
                     await sock.connect(listener.getsockname())
             assert sock.fileno() == -1
 
+@slow
+async def test_SocketType_connect_paths_not_listening():
     # Failed connect (hopefully after raising BlockingIOError)
     with tsocket.socket() as sock, tsocket.socket() as non_listener:
         # Claim an unused port