If you are mainly asking about why banks fail and whether they can fail in the situation you describe, you are absolutely right.
Banks do not keep their depositors' money on hand in their vaults. They lend it out so they can make money. If all of a bank's depositors demanded their money at the same time, the bank would fail because it could not get enough money to pay them all.
Because of that, our banking system is based largely on confidence and trust. That is one reason the Congress created the FDIC back in the Depression -- so that no one would need to worry about losing their money if their bank went broke.