If you think a bank run is going to happen, you want to withdraw your money ASAP, before it's all gone. Everyone else does the same - so you can get a self-fulfilling prophecy, even if everything would have been fine if everyone had stayed calm. For personal banking, this is largely mitigated by FDIC insurance, because the Fed can print enough money to cover small bank failures without anyone noticing or caring. However, if you're fucking around with collateralized debt obligations (CDOs)...2008 happens.
Banks actually pay insurance premiums to FDIC, which are held in a fund which is used to pay claims. If the fund is wiped out, practically speaking the Fed would cover it but that's not the normal method of operation.
The "Calculated Risk" blog reports on bank failure, and even in 2010 (11?) when literally hundreds of banks failed there were only a couple cases in which the FDIC had to pay anything out.
which states that "The fund had a balance of negative $7.4 billion as of Dec. 31, though that was an improvement from the $20 billion hole it was in at the end of 2009."
