Sounds like something killed the custom capability for Connections that are added to the Roles? Can you access the Connections : Roles admin page? If you can, try resetting all role capabilities. If you can not access the roles admin page, which I did not think you will be able to if the capabilities have been removed…
Install the Code Snippets plugin.
Download and unzip this file.
Import the code snippet XML file, search.code-snippets.xml, into Code Snippets. See the FAQs on where to find the import feature.
Activate the newly imported snippet.
It seems that somehow once in awhile the custom capabilities for Connection are corrupted or removed. This Snippet will reset them. Activate this Snippet and verify that you can access the Connections admin pages. Then deactivate this Snippet. If you do not deactivate this Snippet, the role capabilities will be reset on every page load and that would be bad.