The FDIC covers up to $250k for each account. It does so using a fund that's about $50b, for a depositor liability of around $5t. If you do the math, that means around 1% of depositor liabilities are covered. There are over 5000 institutions covered, however JP Morgan Chase holds around $1t in deposits (20%), BofA holds around $2t (40%) and then the rest is divided up in the long tail of decreasing market share.
This means that, if for instance BofA or JP Morgan Chase fails (that'd never happen, right?) that the liabilities alone of these banks are already 20x to 40x bigger than the entire FDIC fund.
Please tell me you see the problem there.