One way this can happen is if either the Share itself, or the folder of the share, don't have the right permissions. Assuming you are on a Domain network, make sure all users have update rights.
Don't try to finesse NTFS permissions. It is more tricky than you may think. As a test, you may want to give Everyone full access (again, to the Share, and separately to the folder), then gradually back off. Be sure to test with 2 users.
-Tom. Microsoft Access MVP
