GM-GMC: Starting problem with Jimmy, jumper cables, starter solenoid

Question
I have a 2000 Jimmy with 83,000 miles.  Every two months or so the car refuses to start.  The key will turn and the engine lights will light but for some reason the engine will not turn over.  This problem seems to fix itself after 2-3 hours of sitting idle (or after the car is towed to the repair shop).  Any attempt to jump start the car with jumper cables does not work. The ignition cylinder and the battery was replaced by the dealer but the problem persists.  Any ideas?  

Answer
These are the kinds of problems that drive everyone crazy!

Did the dealer replace the cylinder or the ignition switch?  The cylinder is where the key goes.  The switch is down the steering column near the firewall and is operated by a rod from the cylinder.  The switch, if it was not replaced, could be going bad.

The other issues involve corrosion of battery cables and ground wires.  The ends where the cables attached should be removed and checked for corrosion.  Also, cables will corrode under their insulation and cause these types of problems.

The starter solenoid is also a potential source of trouble.  They sometimes will not work properly when the engine is warm but start the engine fine when it is cold.

I would make sure that the ignition switch was replaced and then check the cables.  The starter (and solenoid) can be removed and checked by an auto parts store, but it might test good since it will have cooled off before it is tested.