It depends on how you have your money in the bank. If it is in safe deposit boxes, then you should be safe from any bank run.
I hope you're kidding. If it was this simple, there would not have been a Great Depression. The danger is devaluation of the dollar, meaning if FDIC goes bust and is bailed out by firing up the presses you'll need a wheelbarrow full of dollars to buy a loaf of bread. If that happens it won't matter if your cash is in your mattress or in the bank--it will all turn into monopoly money overnight. The only reason to withdraw cash at this point would be to hold you over for a couple of weeks until the money, worthless or not, is put back in your account. The trillion dollars they are apparently going to throw at this problem next is going to have essentially the same effect, but instead of printing the money we're going to borrow it, probably from the same people that already own these worthless securities, i.e. we're borrowing more from them so we can pay them back. It's pretty insane.
Then FDIC was enacted in 1933... the Great Depression was for all intents and purposes, over.... the majority of the damage had already been done. If you are young, count on the $USD being devalued to a minimum of 10... vs currently 77... during your lifetime... But in the mean time, currently insured FDIC deposits are not at risk. But if you young people hope to have any kind of future in America, you should be BITCHING YOUR HEADS OFF AND STORMING THE WHITE HOUSE...
He already understands that printing more money devalues the dollar, and fwiw "printing money" is used loosely on the boards to mean both literally printing money and in a figurative sense, increasing the money supply by borrowing, lending, etc.
They don't really even have to bother with the "printing" part... just create a credit entry with the stroke of a key... And to think... at one time they had to "clip coins and melt down the scraps" to inflate.... What a hassle! Those times must have really been CRAP!
Keeping bank accounts at small regional banks, with more than the FDIC limit is utter insanity in this environment. Even below the FDIC limit, it might run out of funds and there could be delays while they get authorisation for more funding. I would recommend keeping your cash in T-bills, and a spread of "too big to fail" conservatively managed banks without massive CDO/sub-prime/housing exposure like J P Morgan, Wells Fargo, US Bancorp etc. It would also make sense to have some bank account(s) abroad in safe countries. In a true banking crisis, safe deposit boxes will be sealed like in 1933 under FDR, and you won't be able to get your cash/belongings out. Keep some cash at home and other trusted locations, in a fireproof safe. Gold is also worth having in small amounts purely as a crisis hedge - and many people are happy to be long it for fundamental reasons too.
FDIC and the US has unlimited amounts of money... Their credit is AAA and the US economy is "fundamentally strong" - listen to the politicians. Don't worry about a thing - they know what they are doing, and they are always in your corner - looking out for your money.
Here is an interesting article... 5 Banks Safe from the Storm http://articles.moneycentral.msn.com/Investing/SuperModels/5BanksSafeFromTheStorm.aspx?page=1
One would be wise to buy some hard assets that the feds cannot inflate to zilch. Gold is already nuts, any other ideas?