Have the users who are getting the log in pop up opening their database with an account name/password that also exists in the other file?
When a script or relationship refers to another database file, it will attempt to open it with the same account name/password as what was used to open the current file. If it matches, no password pop up. If it doesn't, you get the request for a password.
If you are sure that you have matching account names and passwords in both files, recover both files ASAP and see if the recovered files show the same problem. I have encountered unexpected requests for a password when one of the files was damaged.