Any bank could loan out an infinite amount of money. If you come to a bank and ask for a loan (and they approve), the bank will create the loan (an asset on their balance sheet) and a matching deposit (a liability on their balance sheet). They could do this an unlimited number of times, as long as the deposits they create are recognized and accepted by the economy. Generally banks will have some reserves (currency, accounts they can draw on, etc) to pay out people when they withdraw from their deposits, at their discretion

The problem comes if people try to withdraw or move their deposits, and the bank does not have reserves to cover the withdrawal. In these cases, the bank would need to sell some of it's assets -or- borrow money from somewhere else. If they can't liquidate assets fast enough, they have a liquidity crunch and will need to borrow or sell the company. If the value of the assets is below the amount being withdrawn, they're insolvent.

The Fed can act as a lender of last resort in liquidity times, but if a bank is insolvent they likely won't be able to post collateral to get loans from the Fed.