Thanks to all who have posted about the resistor and low/high speed fan problems.
Now I have one that has me totally stumped:
Let me start by giving the basics - 06 Cooper S HardTop. Stop and go traffic with the A/C on and car started overheating per the little red temp light. Turned off the A/C and traffic quickly cleared and temp came down quickly.
then it did not happen again over a couple of days.
sat with the car idling in 110deg F with A/C on for an hour... no problem. Went driving for another 30 min, slow and higher RPMs - still no problem.
Stopped at a place and let it idle again - nothing, nada!
pulled into a gas station and was going to wash my windscreen, BANG temp started rising... checked, fan was not turning. Turned off the A/C and got some speed/airflow returning home real quick.
Using all your collective guidance, i starting testing things out.
So i unplugged the connection to the fan on the right front corner. using gator clips and wire I manually grounded and applied power to the fan - Nothing.
walked away for about 15min. Came back and applied the manual power again and zoom... fan is fully functional.
walked away again and came back to apply power again and nothing happened. i confirmed that i had power by striking the hot and ground together resulting to sparks.
testing again and no action from the fan. proceeded to walk away again and come back and apply manual power again and wala, the fan works both high and low.
Now I cannot get it to fail again.

I have it out the car now waiting on a replacement fan, but i cannot get it to fail again.

Any ideas

 

Intermittent electrical connections are a $itch. Obviously a janky connection in there somewhere. I once had a wire that was broken inside the sheathing. Sometimes it worked, sometimes not. Finally I messed with it enough that the broken piece came out and I realized it was broken.
Throw in the new fan and hope it's on that side of the connection - likely so.