The term that you are looking for is "bank run" or "run on a bank." This is the term for what happens when a large number of depositors all try to take their money out of a given bank at once. This is a problem if so many depositors try to take money out that the bank runs out of cash to give them.
Banks do not simply keep all their deposits in a vault. Instead, they loan the funds out and only keep small amounts of liquid cash around. They rely on the idea that people are not all going to want their money at once. But if all the people do want their money at once, the banks are in trouble. This is what happened at the start of the Great Depression. People worried that their banks would fail so they started runs on the banks. These bank runs ended up making many banks fail in the early 1930s.