My neon has this trouble off and on. Starter will not crank over. I had the battery tested and they said it was good. Have cleaned all the battery connections. sometimes the dome light and radio work when I have the starting trouble and sometimes they don't. I car hear a click sometimes when trying to start it but it seems like it is in the area of the battery or ecm. Sometimes I can hear a click from the starter and a few times the starter would spin without engaging but has not did that for along time. A few times when driving the car the radio would quit playing and the guages would die and the fasten seatbelt light would come on and a chime or something would sound, that only lasts a few seconds. That last trouble has not happened since i tightened the nut on the cable that runs from the positve battery post to I believe the starter.
Anyway to get it to start, I either jump start it or try to tighten that nut on the postive battery post for the cable that runs to the starter. (that nut is very tight already). I have cleaned all the battery connections and posts and they are very tight. The battery does not seem dead at all because like i said sometimes just turning that nut makes it so the car will start.
I have tried also moving the gear shifter back and forth and starting it in neutral but that does not seem to have any effect. I have also noticed that when the starter will not crank over and the dome light is on that the dome light will dim and go out when the key is turned to the position where the car should be crancking over.