While the FDIC fully insures deposits up to $100,000 and up to $250,000 in IRAs, working with a bank that is going under may still be difficult.  For example, checks that you write from a failed bank may not clear for a certain time period.  If you require access to your account to be uninterrupted, it is best to move your entire account to safer banks before they fail.

Also, the balance sheets of the Federal Reserve and the FDIC are only so large.  They are not setup to withstand fundamental problems with the banking system (such as the larger portion of the banking industry taking on far too much risk).  If we get enough bank failures, all deposits of failed banks may be lost or the losses will be bailed out by the government with new money which will make inflation worse.

Examples of recent bank failures
During the week of July 16, 2007, The Bear Stearns Companies, Inc. (one of the largest global investment banks, securities trading, and brokerage firms) disclosed that two subprime hedge funds had lost a significant portion of their value because no one would buy the funds unless it was at a deep discount.  Because of accounting rules, the price that the securities sold at determined the value of all the remaining securities of the same kind on Bear Stearns' balance sheet.  The Federal Reserve issued a non-recourse loan to JP Morgan Chase.  The Federal Reserve effectively assumed the risk of Bear Stearns' less liquid (sell-able) assets.  This isolates JP Morgan Chase from taking losses beyond those of Bear Stearns' assets.  On May 29, 2008, the sale to JPMorgan Chase was approved.  The unusual Federal Reserve non-recourse loan was necessary for JP Morgan Chase to take on the risk of buying Bear Stearns.  This is because if this bank defaulted on its obligations, it could have started cascading bank failures due to other large banks dependent on Bear Stearns.

On September 12, 2007, Northern Rock (UK's fifth largest mortgage lender) asked the Bank of England for support.  Although it had sufficient assets, it was low on cash and lenders did not lend enough cash to Northern Rock due to fears about the subprime crisis in the United States.  On February 22, 2008, Northern Rock was nationalized because there were no successful bids to take over the bank.

On May 12, 2008, IndyMac Bank, FSB (the seventh largest mortgage originator in the US) reported in its 10-Q (a quarterly report to the SEC) that its risk-based capital ratio had dropped to 10.26% (10% is the minimum) due to a significant increase in loans that are in default (roughly, loans with payments past 90 days due).  On July 11, 2008, the bank failed and was taken over by the FDIC.

More bank failures likely
While I do not want to go into my understanding of the economy in this topic, I do have a reasonable estimate that we are only one third into a depression based on credit deflation.  The FDIC keeps a list of banks that are showing signs of definite weakness.  As of late May, 90 institutions were on that list.  I think we are going to see more bank failures, possibly including some large banks.  The problem is not just with subprime loans.  Other forms of debt are at risk of having higher rates of defaults: prime mortgages, automobile loans, credit card debt, student loans, etc.  The rate of new defaults are going to get worse as adjustable rate mortgages continue to reset, the cost of living (gas, oil, food) continues to rise faster than income, and the unemployment rate gets worse.

What you can do
As you can see from the dates in the examples section, there seems to be a long delay from warning signs to failure.  I do not always expect that to be the case, but it is encouraging to know that, for now, people are likely to have time to move to safer banks if they act on the warning signs.

There are companies that rate banks.  They are listed here:
http://www.fdic.gov/bank/individual/bank/index.html

Note that not all of them may be useful for rating the risks in the current environment or have up-to-date data.  Here are a couple of rating services that are free:
http://www.thestreet.com/tsc/ratings/screener.html
http://www.bankrate.com/brm/safesound/ss_home.asp

For example, as of July 21, 2008, bankrate.com has a memorandum on IndyMac Bank that states: "Excessive loan yield can also be an indicator of existing or future credit problems, and our loan review suggests that the institution has assumed a seemingly prudent position between credit risk and financial reward."  This seems to be based on data from March 31, 2008.

Another way to estimate the risk is to watch for a sudden drop in the bank's stock price.  For example, before IndyMac Bank's last 10-Q report, it closed at $3.43.  A month later, it closed at $1.72.  When the stock drops below $1.00 for a certain period of time, it becomes delisted from the main stock exchange, so any bank that has a share price below $2.00 indicates trouble.  On June 26, 2008, IndyMac Bank's share price dropped below $1.00.  However, comparing share prices between banks does not give a useful safety comparison.

Banks typically limit how much cash you can withdrawal in one day, so do not expect to be able to take out massive amounts of cash.  You may be able to raise that limit by giving them advance notice (you may be required to give a 7 day advance notice).  Better ways to move money between banks include writing a check to yourself from the source bank and depositing the check at the destination bank or using a wire transfer (there is usually a $20 fee involved with wire transfers).  It helps to be ahead of the crowd.