You have a legal contract with FDIC which insures all deposits upto $250K is what I know. If the bank goes bust FDIC owes you your money under 250K.
You are right in practice that the FDIC guarantees small deposits.
If it was just a legal contract, in the worst case you'd have to sue to hold them to account.
(On the other hand, politics can change from one day to the next; but contracts are harder to unilaterally change.)