It is just as @pele23 states.

There isn't enough paper notes in existence to withstand any kind of surge in people withdrawing funds.
So, if the banks did nothing, the ATMs would quickly run out and the banks would have to honestly say they had no notes to give out. Of course, all of your money is still safe inside the bank, they would say.

But, really, the govern-cement gets involved rather quickly and imposes maximum withdraw limits.
Lines for ATMs would become rather long, rather quickly.

In the end, the sad thing is that no banks would collapse. People see it as a crisis caused by people or circumstances. They do not see that the problem is the banking system.