Hmmm... I hadn't considered that exact question, although I suppose it is a possibiltiy with any account as all now exist in digital form on a bank's system somewhere. I am not sure what the rep could have meant by your "being responsible", I think that banking regulations prevent you being held responsible for fraud against the bank (the bank's system is hacked), but maybe I don't understand what you asking.
Mr.Tours and I did do basic research about BankDirect, verifying it's FDIC coverage and checking the stock performance of the parent bank.