Error when editing SharePoint List

You might get the following error message when editing SharePoint List Items:
The server was unable to save the form at this time. Please try again.

Analysis

Using Fiddler, I was able to track the issue down to the following request of the CSOM API:

/_vti_bin/client.svc/ProcessQuery

Fiddler listed the following error messages:

ErrorMessage=Access denied. You do not have permission to perform this action or access this resource.
ErrorTypeName=System.UnauthorizedAccessException

SharePoint Configuration

The users experiencing the issue had contribute permissions set directly to the affected list.
Within the SharePoint Site, no permissions were granted to the users.
They were able to view the items, however they got the error message when editing existing ones or when creating new items.

Resultion

Make sure the users have at least the permission
“Use Remote Interfaces — Use SOAP, Web DAV, the Client Object Model or SharePoint Designer interfaces to access the Web site.”
set on the Site level.

You can also verify that the users are able to browse all lists using CSOM:
/_vti_bin/Client.svc/web/lists

If they don not, they will get an access denied here as well.