In what country? Certainly not the US for FDIC insured accounts. My bank went under in 2009 and I never lost access to money.
https://www.fdic.gov/deposit/deposits/
Ownership category is just something like single, joint, retirement, etc. If you have more than $250,000 spread across multiple accounts but with the same account holder and ownership category, the excess is not covered.
Let's also make a note that people with $250k+ in cash savings are a tiny group that should have no really bearing on whether cash is trustworthy.