Just to clarify, an EPA is only valid while both parties are alive, so if you were to pass away it becomes useless. It would really come down to the individual you dealt with at the bank as to whether or not they took the sensible road, as opposed to the "letter of the law" road.

Having said that, in this day of electronic everything, you would still find it pretty easy to access funds. One of the executor's jobs is to notify banks of the account holder being deceased, so until they're told by your executor they wouldn't know. Once the account holder passes away the account should be frozen until probate is granted. If an account is in joint names, then the survivor can continue to use the account as normal.