If you're dealing with linked tables, you can also try deleting/relinking the tables. When you link/refresh tables, there is a "Save Password" option which it sounds like got check/coded somehow. It would then no longer prompt you for a password. If this is the case, you can delete/relink the linked tables and should then get prompted for a password.
Otherwise if the password is related to the backend/JDBC connection, you'll need to look there to troubleshoot. I will sometimes delete my ODBC DSN file and re-create it in which case you have to supply a password (if required). I'm not sure how it works with JDBC.
If the mdb is protected via an mdw file (ie. uses MSAccess Security), you can use this utility to view the mdw file logins/passwords. I believe it ONLY works on *.mdw files though.
Expert Database Programming
MSAccess since 1.0, SQL Server since 6.5, Visual Basic (5.0, 6.0)