Banks lend deposits. That's how they make their money. They have to keep some part of the deposits as a reserve.
Real regulated banks have access to the Fed to borrow in a case of a bank run.
Real regulated banks have access to the Fed to borrow in a case of a bank run.
Anyone who has more than the insured amount, or anyone who needs their money in the near future will still engage in a run.
There were several runs during the 2008 crisis, the most famous is the run on IndyMac Bank.