I have a 91 3.1 Camaro. Ever since I've had the car, it acts like the starter solenoid is bad. You turn the key, sometimes it just clicks then finally it'll engage the starter. I've replaced the solenoid and the starter and still have the problem.
Recently, I had the engine out of the car so I did some testing on the starter with it in the car. Every single time you hit the key, the starter worked. Once I put the engine back in, I was having problems again.
Here's my thoughts... possibly still a solenoid or starter problem but not likely. Possibly a bad wire somewhere that was either making better contact the way the starter was position without the engine in OR just a bad wire the once it has the load of trying to turn the engine over, won't carry the amperage. Any thoughts or experiences? I'm getting really tired of this problem! Thanks!

 

I'm currently having the same problem. Mine will only do it sometimes, maybe 1 out of 10 starts. I can't get it to do it on command, so it's a pain to try and troubleshoot. I'm thinking it might be the VATS starter enable relay, I'm going to try replacing that.

 

Clicking shows the solenoid is working but not turning over means the starter is not getting enough power.
Check batt connections and the main ground wire from batt to engine for loose / bad connections ( assuming your batt is good)

 

If you get any activity ( clicking ) from the starter is not VATS related because the start enable relay has to be working for the power to get to the solenoid.
If VATS related , turning the key to start gets you

 

Yeah, I had wondered about VATS but had ruled it out do to what vetteoz said.
As far as the battery and possibility of loose connections...I've had atleast 2 batts in it and have had all of the connections off and back on a few times.
I appreciate the ideas, though! Keep them coming, please!