Electrical problems?

I was driving down the road. First the radio went to static, then the dash lights went out. As I continued driving the headlights kept getting dimmer. The engine started to lose power. I pulled over to the side of the road, cut the engine off. Of course it didn't start back. Assuming it was the alternator, the next day I went to jump it. Nothing. After a few minutes of letting the battery charge the wipers, starter and a fan (I'm assuming) all engaged. Took the battery home and completely charged it. It's a brand new battery. When connecting the battery, everything engaged again. My friend's dad thinks a solenoid grounded out in the starter. I'm not convinced. Any help, ideas would me much appreciated! Thanks in advance.