02 VW Jetta 5 speed won't start-reset battery works

Hey guys, question for the experts or vehicle
adepts. I recently bought an 02 Jetta 5speed for
my mom. Drove it fine for... I'd say 4 days,
managed to make a 200 Mile trip to see my
grandmother. Then, the car stalled out, EPC light
came on. My mom got the car to restart, just the
CEL light was on. We brought it back to the dealer
we got it from, they had it for over a week. They
said the fixed the problem, and we picked it up 2
days ago. Ran fine for a whole day. Then we had a
new problem, and I can't figure out what it could
even be.

It didn't start. Engine kept turning, it just wouldn't
start. My mom got it to start after letting it sit for
10 minutes. She drove to work, no problems until
she gets ready to leave and... Didn't start again.
She gets it to start after... Ten minutes or so, drives
home, so I take a look at it. Doesn't start AGAIN. So
I give it a few minutes, still no start. I pull the
battery out, plug it into a charger, battery is full.
Plug battery back in, it runs fine. I think "well that's
odd, let's give it a test drive." Drives fine. We park,
kill the engine, and out of curiosity try to stay it
again. Engine just kept turning, no ignition. I try out
a hunch, disconnect the battery and reconnect it,
car runs fine.

Tried again this morning aaaaaaaand car won't
start, engine keeps turning. I tried jumping it
aaaaand... Yeah no dice. Resetting the battery
works fine though. Any idea what it could be? The
CEL is on someone's when it starts, other times it
isn't.