Likely its a defective part...
No-start and no-crank problems aren't the easiest to troubleshoot. You
didn't give us enough information.
- Does your car crank but not start, click but doesn't crank, doesn't
do anything... when you turn the key the first time(s)?
- Have you tried jumping it or have you always just tried for several
minutes then it started? Don't just tow, try jumping it.
Just need a lot more information about the specific details on this
problem. Off hand it sounds like it could possibly be a low battery, a
bad cell in your battery, or a defective starter.
- Is your Neon Auto or Manual??
- If the dealership told you there wasn't a problem, they MUST haven't
had problems starting it because if they did they would've tried to get
you in for service (warranty or not). That means it COULD be user
error. Make sure you use the right key every time, make sure the car
is in P or N every time you try to start it. Make sure you have the
brake on Auto, or the brake and clutch depressed when you try to start
on a Standard.
That's the big thing, I guess. If the dealer said there was no
problem, there might not really be one.... But LOL at that, because I
they've been wrong before. Get on your local Craigslist you can find
someone to come check your car out for dirt cheap.