Yes you can, as long you have a valid SSN that match your information, you can get a Credit Card. If you never had a CC before, you may got denied with some banks because you don't have a credit history.

The Government will provide you a SSN as long you have VISA/Green Card etc, in case you don't qualify for a SSN, you can get a ITIN number, which is valid when you have to do your taxes (a non-legal resident can get an ITIN number without any problem).

The only limitation is... you can not work with out a work permit (employment authorization card), but for credit cards/bank accounts, you can get most of them without any problem. Even Chase and BoA will open a Checking Account with a valid passport.