Panic demand. Overwhelming requests for resources. A common occurrence during the Great Depression of the 1930s. Some people with money in a bank try to withdraw their savings out at the same time in fear of bank failure; no bank has sufficient funds for all accounts to be liquidated at the same time, so the bank does, indeed, go bankrupt; often leaving many without getting their money back. So this phrase is used whenever a supply system is overwhelmed. (Compare to Suck all the air out of or Up to one’s neck in crocodiles.)

“When I tried to buy a hot dog, they were all gone, he said they had a run on the bank during lunchtime.”