Because of fractional reserve banking. 90% of the money in your bank account are in fact credit created by the private banks out of thin air. If the banks fail, 90% of the money in the entire system will suddenly disappear. Imagine that.
The loss of money in the commercial sector means (after a series of cascading failures) we're all unemployed and burning dollars to stay warm.
Close, it's actually $250K per depositor per ownership category per institution. Multiple accounts in the same category don't add insurance limits.