Yes, you can pay as many of your debts as you want or need to pay after you file bankruptcy. Certainly, if you want to keep your house and car you must continue paying the lender. You are also obligated to pay the debts that for which you enter into a Reaffirmation Agreement. (See discussion above on reaffirmation agreements).
Do not take on any new credit for at least one year after bankruptcy.
I recommend that you not make any payments that are not absolutely necessary. My clients tell me they have more money than they ever thought, once they were relieved of the obligation to pay the high interest rates for their credit cards and other unsecured debts.