I do not work in IT nor do I claim to be an expert.. but my brother does and from my understanding, it is very much like a kite. Sometimes the kite will not pick up and fly through the air because it is not in the right wind, right? You can sometimes get a lot of speed up and the kite will fly.
If you take off the casing of the laptop (as in, reveal the fan), you should be able to get the fan working again with very little trouble. The manual way is to blow on it until it picks up and registers movement.. and should restart. Sometimes you have to do it for a good 30 seconds to 2 minutes.. it has never taken me longer than that.. but I would encourage you to try for four minutes maximum and then take it to a computer shop. Please try this and edit your post with your results - I hope it works. :)
Once it happened to me when I did not have enough money to take my computer to the computer shop.