Car Battery Won't Hold Charge?

I have checked the water in the battery. The cable are new, (and so is the battery). When I drive the car for longer than 5-10 minutes it dies. I jump start it for a few minutes and it will run for another 5-10 minutes then dies again. Someone told me they didn't think it would be the alternator because if it starts, it should continue to run because the running itself would keep it going. The car will also reset the radio station to an AM station if I open a door while it is running, or if I turn car off. Also, the rear window wiper will start every once in a while just out of the blue.

Suggestion:

Have your alternator checked. If the alternator is bad the car will run for a few minutes until the battery goes down too far to keep it running. If the alternator were good it would continue to run even if the battery was disconnected.