Do you hear a single click or clonk sound when you turn the key to "start" position and the car does not start? If you do, it is most likely the contacts in your starter that are worn out. Common problem. The second fan (I assume you refer to the one on the radiator) could be just a coincidence, if the car is warm it will come on as soon as you turn the ignition on. Then by the time you have rocked the car back and forth, it would have cooled the fluid in the radiator and thus shut down.
From my experience when the starter contacts are getting worn the car usually starts after a couple of tries, like 5-15. Then it works fine for a couple of days, and then it fails to start again. Getting progressively worse.
Getting the starter out is a bit awkward, but once you have it out, replacing the contacts is cheap and easy.
Check out this thread for instructions:
Repair your own STARTER