You have no 'obligation' other than that of customer service. Anyone, of course, can claim that someone gave them a particular bill - how can that ever be proved. It's generally less likely that it came from a bank, because (hopefully) all currency received and processed through a vault are ran through a sorter/counter that has counterfeit detection technology.
Now, it's an excellent idea to examine the value of the custoemer, believability, etc in making a determination on assisting them with any reimbursement.
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My comments are absolutely no reflection of, nor influenced by, my employer - take them at your own risk.