Some light has been shed on the 'mystery' in the previous post. The Federal Deposit Insurance Corporation, which is in charge of exactly what it sounds like, was unable to collect insurance premiums for 10 years (1996 - 2006).
From what I can tell - the news is new to me - banks are required to pay 1.15% of deposits as premiums to the FDIC. Unfortunately, the FDIC wasn't actually given any power to collect the premiums with.
As a result, while the banks were doing well, they refused to pay the premiums, and the FDIC's holdings fell to about 0.4% of insured deposits.
Once they were no longer doing well... that was a little too late to start paying.
So, YES, given this situation, a sharp drop in the reserve ratio could very well cause a bank run with no safety net.
Also, YES, this is as silly as it sounds. As a Consumerist commenter put it, "I kind of like this logic. I mean, I've paid homeowners insurance for a good long time and it hasn't caught on fire so far--so they should just be content with the money they have collected and continue to cover my risk free of charge."