As of yesterday I got my automatic '86 back from the shop because the starter was going out. It started up fine two times, but then completely quit. It was clicking, so the solenoid was engaging, but no matter what it would not start. We ended up having to jump start it, in which case it started right away. We got both the battery and alternator tested and both were fine. We are replacing the alternator tomorrow though since a bracket snapped off.

Any clue as to what this is and how I can fix it?
Thanks!

 

If the battery is dying, you're looking at a charging problem / bad battery / battery drain (short).

If the battery is charged, but you turn the key and only hear a "click", you may have a problem with the start circuit: https://www.rx7club.com/2nd-generation-specific-1986-1992-17/ive-solves-2nd-gen-sticky-starter-mystery-896013/. The ignition switch or interlock switch could be to blame, or just an overall buildup of resistance (which is common after 25 years).

 

The battery wasn't dead which was the weird part. All my gauges started working, my headlights were fine. The car didn't "click, click vroom" like in the other post. It "click, click, click, click, NOPE." I just bought a brand new battery about 5 weeks ago.
How can I go about checking the ignition switch and the interlock switch?

 

I'd suspect a bad ground at the harness side.
The Harness goes through the top bolt of the starter at the engine and it can be tight but not enough to let the starter gets it's full power.
Since you just changed the starter that leads me to believe that you didn't get good contact.