In Great Depression-era America, before the era of online banking, there might be only one choice of bank for people living in a small town. So when a bank was revealed to be undercapitalized, there would be a literal "run on the bank" where people raced to turn their deposits into cash before the bank ran out of cash.
If there is only one bank, and it does not have the ability to handle all depositor withdrawals, via backing from the money-creating sovereign, there will be a bank run.
This is where the other answer's comparison to Soviet Sberbank breaks down. There was never any possibility that deposits in Sberbank had counterparty risk. Because it was the state bank, any shortfall in assets would surely be covered up with money creation.