Large investment banks/commercial banks of the kind you read about that needed bailouts during the real estate crash and subsequent market crash are susceptible to cash flow squeezes during times of market turmoil. This is because they have huge amounts of money invested around the world and they get tied up with loans to countries. This sovereign debt is also tenuous - Portugal, Italy etc., etc. To what extent these investments have been sold, bought by the fed, bought by the EU central bank, nobody will ever know.
As a an ordinary citizen, it is my belief that there is no reason to be exposed to that risk if all you want is a checking account, a savings account, a credit card, or self-directed IRA.
Most investment counselors recommend that you keep your money in a regional bank with no exposure to the things I have mentioned. Most regional banks don't have those exposures. There are even smaller banks that are local with very good balance sheets. You can check on the status of these banks on the Internet.
I would suggest going to http://banking.about.com/gi/o.htm?zi=1/XJ&zTi=1&sdn=banking&cdn=money&tm=20&f=00&su=p284.13.342.ip_&tt=2&bt=0&bts=0&zu=http%3A//www.bankrate.com/brm/safesound/ss_home.asp
There you can put in your state, check star rating (I would put in 4 stars), and select 1-5 billion dollars. This will give you the size of bank you probably want to look for. At that point Google the bank for what analysts think and go to the bank's home page. You should see a pattern of positive or negative ratings.
I bank in my hometown with a regional bank that has a very high rating - these banks probably will have less problem and avoid shutting doors in the event a big financial crisis.