Are you talking to ppl with >250k in the bank? Cause' it would take a pretty serious incident (not just a run on banks) for FDIC insurance not to kick in.
If you're truly concerned about a bank run - break up funds and open accounts at multiple major banking institutions (both local credit unions and major banks). If needed, one can pull ~$500 per ATM/bank per day.
No need to panic.
Vox populi,
"Facts do not cease to exist because they are ignored." -A.Huxley