On my basement unit the #1 compressor would not run. After a few seconds the #2 compressor would come on and run fine.
I removed the control box cover and the problem became apparent. I saw a burned wire in the #1 compressor relay. The jumper wire was burned completely off one end. Looking closer the same jumper wire on #2 compressor has been really hot also.
These wires have crimped on terminals that push on the relay terminals. They must have been a little loose and caused the heat. I replaced both jumpers and everything fine now.
Could have caused a fire but thankfully didn't.