Banks and, at least VISA, are not as quick to do chargebacks they used to be. Now, they'll only do them if the customer says their card was stolen. Found this out when my mother was scammed by a moving company for the 50% deposit (company never showed). She called the number on the back of the card to dispute the charges (I believe its the bank's customer service number, not VISA's). But the bank told her they couldn't reverse the charges since she was the one who authorized them. To get her money back, she'll have to take the moving company to small claims court.
I have used AMEX since the last 9 years, precisely for things like these. They are REALLY good at processing disputes and chargebacks, and in the last 9 years, there has never been a time when I 'lost' a claim against such fraudulent charges (I've had around 5 such claims).
On the other hand, both Visa and Mastercard are pretty much useless to 'protect' you against charges like these.
Debit cards (which from the way you're talking about a bank it sounds like this was) are like that. It's why your best practice is to always use a credit card when buying things (it's not like it costs you anything, as long as you have the discipline to not run up a bill you can't pay).