Yes, just like a non-FDIC insured bank can tell you 'too bad, too sad' if they're robbed and your money was stolen. Unless you're specifically insured against loss, banks aren't obliged to give you money lost due to robbery unless you some how prove the robbery was the result of, say, gross negligence.
Yes, some state (rather than federal) banks are not FDIC insured. (Also, credit unions are a lot like banks and are not FDIC insured, but most of them are CUA insured, which is functionally the same thing.)
