While I doubt there's a server in each branch managing account balances, there's is some kind of branch-related weirdness involved in bank account management at some banks.

With some banks, if you open a bank account at a branch in one time zone but later move to another time zone, your transactions will still be processed according to the business hours where the account was opened.

I know someone who is in this situation and they can't make transactions for several hours towards the end of the day (after midnight in the originating branch's timezone) because their bank's online banking front end can't agree with the bank's backend system about what day it is.