If there is a change of address with the PO, then ALL of the mail would be forwarded. After a few days, if the person is still living at the home, they might question that something is wrong.
If there is a legitimate change of address, and the bank mails one notice each to the old and new addresses, the customer will be getting the one for the new address first, followed closely by the forwarded envelope.
You may want to contact your local postmaster or postal inspector to see how address changes are handled locally.
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Integrity. With it, nothing else matters. Without it, nothing else matters.