--- a/devtools/client/webconsole/new-console-output/test/mochitest/browser.ini
+++ b/devtools/client/webconsole/new-console-output/test/mochitest/browser.ini
@@ -53,18 +53,16 @@ support-files =
   test-bug-595934-workers.html
   test-bug-595934-workers.js
   test-bug-597136-external-script-errors.html
   test-bug-597136-external-script-errors.js
   test-bug-597756-reopen-closed-tab.html
   test-bug-599725-response-headers.sjs
   test-bug-601177-log-levels.html
   test-bug-601177-log-levels.js
-  test-bug-603750-websocket.html
-  test-bug-603750-websocket.js
   test-cd-iframe-child.html
   test-cd-iframe-parent.html
   test-console-api-iframe.html
   test-bug-618078-network-exceptions.html
   test-bug-630733-response-redirect-headers.sjs
   test-bug-632275-getters.html
   test-bug-632347-iterators-generators.html
   test-bug-644419-log-limits.html
@@ -155,16 +153,18 @@ support-files =
   test-result-format-as-string.html
   test-sourcemap-error-01.html
   test-sourcemap-error-01.js
   test-sourcemap-error-02.html
   test-sourcemap-error-02.js
   test-stacktrace-location-debugger-link.html
   test-trackingprotection-securityerrors.html
   test-webconsole-error-observer.html
+  test-websocket.html
+  test-websocket.js
   testscript.js
   !/devtools/client/netmonitor/test/sjs_cors-test-server.sjs
   !/image/test/mochitest/blue.png
   !/devtools/client/framework/test/shared-head.js
   !/devtools/client/netmonitor/test/shared-head.js
 [browser_console.js]
 skip-if = true # Bug 1406060
 [browser_console_addonsdk_loader_exception.js]
@@ -436,10 +436,9 @@ skip-if = true #	Bug 1403196
 skip-if = true #	Bug 1403200
 [browser_webconsole_variables_view_while_debugging_and_inspecting.js]
 skip-if = true #	Bug 1403205
 [browser_webconsole_view_source.js]
 [browser_webconsole_violation.js]
 skip-if = true #	Bug 1405245
 # old console skip-if = e10s && (os == 'win') # Bug 1264955
 [browser_webconsole_warn_about_replaced_api.js]
-[browser_webconsole_websocket.js]
-skip-if = true # Bug 1408950
\ No newline at end of file
+[browser_webconsole_websocket.js]
\ No newline at end of file

--- a/devtools/client/webconsole/new-console-output/test/mochitest/browser_webconsole_websocket.js
+++ b/devtools/client/webconsole/new-console-output/test/mochitest/browser_webconsole_websocket.js
@@ -1,39 +1,23 @@
 /* -*- indent-tabs-mode: nil; js-indent-level: 2 -*- */
 /* vim: set ft=javascript ts=2 et sw=2 tw=80: */
 /* Any copyright is dedicated to the Public Domain.
  * http://creativecommons.org/publicdomain/zero/1.0/ */
 
 "use strict";
 
-// See Bug 603750.
+// Check that WebSocket connection failure messages are displayed. See Bug 603750.
 
 const TEST_URI = "http://example.com/browser/devtools/client/webconsole/" +
-                 "test/test-bug-603750-websocket.html";
+                 "new-console-output/test/mochitest/test-websocket.html";
 const TEST_URI2 = "data:text/html;charset=utf-8,Web Console test for " +
                   "bug 603750: Web Socket errors";
 
-add_task(function* () {
-  yield loadTab(TEST_URI2);
-
-  let hud = yield openConsole();
+add_task(async function () {
+  const hud = await openNewTabAndConsole(TEST_URI2);
 
   BrowserTestUtils.loadURI(gBrowser.selectedBrowser, TEST_URI);
 
-  yield waitForMessages({
-    webconsole: hud,
-    messages: [
-      {
-        text: "ws://0.0.0.0:81",
-        source: { url: "test-bug-603750-websocket.js" },
-        category: CATEGORY_JS,
-        severity: SEVERITY_ERROR,
-      },
-      {
-        text: "ws://0.0.0.0:82",
-        source: { url: "test-bug-603750-websocket.js" },
-        category: CATEGORY_JS,
-        severity: SEVERITY_ERROR,
-      },
-    ]
-  });
+  await waitFor(() => findMessage(hud, "ws://0.0.0.0:81"));
+  await waitFor(() => findMessage(hud, "ws://0.0.0.0:82"));
+  ok(true, "WebSocket error messages are displayed in the console");
 });

rename from devtools/client/webconsole/new-console-output/test/mochitest/test-bug-603750-websocket.html
rename to devtools/client/webconsole/new-console-output/test/mochitest/test-websocket.html
--- a/devtools/client/webconsole/new-console-output/test/mochitest/test-bug-603750-websocket.html
+++ b/devtools/client/webconsole/new-console-output/test/mochitest/test-websocket.html
@@ -1,14 +1,14 @@
 <!DOCTYPE html>
 <html lang="en">
   <head>
     <meta charset="utf-8">
-    <title>Web Console test for bug 603750 - Web Socket errors</title>
+    <title>Web Console test for Web Socket errors</title>
     <!-- Any copyright is dedicated to the Public Domain.
          http://creativecommons.org/publicdomain/zero/1.0/ -->
   </head>
   <body>
-    <p>Web Console test for bug 595934 - Web Socket errors.</p>
+    <p>Web Console test for Web Socket errors.</p>
     <iframe srcdoc="hello world!"></iframe>
-    <script type="text/javascript" src="test-bug-603750-websocket.js"></script>
+    <script type="text/javascript" src="test-websocket.js"></script>
   </body>
 </html>

rename from devtools/client/webconsole/new-console-output/test/mochitest/test-bug-603750-websocket.js
rename to devtools/client/webconsole/new-console-output/test/mochitest/test-websocket.js