FDIC and state banking departments require every bank to have a formal disaster recovery plan; ideally, one that iincludes real-time data backup at multiple (remote) secure locations. So, once power/telecommunications capability is restored, depositors should be able to access their funds.

On the news, saw a small-town community bank in Mississippi that didn't have power restored, who was allowing accountholders to pull $200 cash out on the "honor system." Due to lack of power/data lines, bank had no way of confirming whether or not depositor actually had sufficient money in their account. Don't expect that from a B of A or Wells Fargo.

Like any other business, effectiveness of back-up/disaster recovery plan varies from bank to bank – some good, some not so good. Hopefully, those in hurricane-prone areas took the task seriously.
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"Things that have never happened before happen all the time." — Scott Sagan, The Limits of Safety