This is describing what happened in 2008. The FDIC limit was only $100,000 but many people assumed they had no reason to worry about the security of higher balances in excess of the FDIC limits. So when big stable banks started to fail or were at risk of failing that caught folks by surprise.

As a result:

> The Emergency Economic Stabilization Act of 2008 is signed on Oct. 3, 2008. This temporarily raises the basic limit of federal deposit insurance coverage from $100,000 to $250,000 per depositor.

The limit was subsequently permanently raised. So with a limit of $250,000 there’s less reason to worry about bank accounts today. But the important point remains, what causes panic is when things that everyone assumes to be safe turn out to no longer be so safe after all.