Types Of Debit Cards

My debit card got rejected at Costco and the CSR told me it only supports certain types (network) of debit cards. Do you know what type of debit cards are accepted at Costco? I'm thinking about opening a bank account, which would help make my decision.

At Costco, you can pay with a check, cash, debit card with your ATM /, and your American Express card. If you use a check, the information must match the information on the membership card. You must use a debit card that has a MasterCard or Visa logo and is not a line of credit attached to it.

At costco.com, you can use VISA and MasterCard credit card and your ATM card and American Express.