There aren't many or any institutions that would send your deposits to someone else, if you weren't financially aware enough to be concerned.

I don't know that any banks warn you specifically when an instrument is not covered by FDIC (when they are trying to sell it to you). It's kind of a scam, because the bank proudly proclaims that it (the bank, in general) is FDIC insured (true) but does not warn you that *not* all deposits nor all instruments are FDIC insured. [forgive all the whacky negatives in the prior sentence - there were actually more of them before I edited it down.] Maybe we need some truth in advertising regulation for banks.