The database is a stand alone, has no relationships to other databases, has no links to free databases, has no plugins that may have passwords?
Duplicate your backup copy, open you backup database copy, import data and test.
OR do a recover of the database (if you do not have a backup), open the recovered database and test.
If the answer to davidanders first question is "no". (ie. your database is related to another FMP database), you can typically avoid the password pop-up if your username and password is the same for both this and the related database.
If username and password is the same, FMP will access the outside Dbase using your first login and won't give you the password popup. When you periodically change your password, make sure you change it on both/all three/etc.
If it's a plug-in or something else...it would depend...