The top 2 FDIC misconceptions that I've encountered include: 1) You do not get the interest earned, only the principal when a bank fails, and 2) It may take years before they release all your funds.See the FDIC's article on the Top 10 FDIC Misconceptions.
If you keep below the insured limits, neither of these are true.
That article also includes links to both the FDIC, which insures accounts in banks, and the NCUA, which insures accounts in credit unions.
It's simple enough to make sure your accounts are under the insured limits. This would be a good time to do that.