We've seen some very significant bank runs in the recent past (Bear Stearns, Washington Mutual, Wachovia), but the deposit insurance mechanism was sufficient to protect the banking system. This won't change without serious shifts in economic fundamentals.
Bank runs are irrational. Had those institutions not suffered a loss of confidence, they probably could have sustained the economic losses and continued in business. Fannie Mae and Freddie Mac are excellent present examples--one of the reasons for the firms' failure was the perceived risk that they might not be able to successfully float debt in the market. At the height of the financial crisis, they needed hundreds of billions of dollars in federal support to prevent defaulting on obligations that would have destructively rippled through the economy. Years later, with the crisis behind, taxpayers are set to earn huge profits from that support.
Economies in general reflect confidence: they're better when people are confident about the future because that encourages them to invest and consume, maximizing the potential economic output; and they're worse when people aren't confident about the future, which retards investment and consumption, reducing employment and erasing opportunities for productive economic output. You could erase all the banks and a lack of confidence would still be just as destructive for the economy.
