Edit your Twitter application at apps.twitter.com/ and under settings ensure "Allow this application to be used to Sign in with Twitter" has been checked. If it is next ensure "Callback URL" and "Website" both link to your sites root ending in / (www.example.com/). If you've caching enabled please disable it and purge it then retest as needed. This typically happens when an access token is attempted to be used more than once.

Seams to be working fine in my tests on 2 different installs with 2 different applications. Once I'm returned to my site I'm presented with the incomplete registration page with the message "Your Twitter sign up is incomplete. Please complete the following.", I supply my email address, click Sign Up and it registers then logs me in fine. So the process appears to still be working fine.

The domain matching is extremely strict on these social sites. So if you visit your site without www., but you configured your application for www. then it likely won't work. To avoid this you'll want an htaccess rewrite or redirect to ensure they stay on the correct domain. Try generating new API keys by editing your Twitter application and under "Keys and Access Tokens" try generating new ones then copy them over to CB Connect.

Aside from the above there's not much I can suggest as I can't review your Twitter application configuration without access to your Twitter account, which we do not like to request.