CPU overheating, the fan won't speed up, but it's not the fan
Solution 1:
I faced a similar problem a while ago. My Acer Aspire 5542G notebook was overheating and the fan just wasn't spinning fast enough to cool down the CPU.
SpeedFan didn't allow me to control my fan speeds so I set out to look for other options and came across RWEverything, which is a very powerful tool that lets you tinker with your motherboard's Embedded Controller (EC) among other things.
Read this informative forum post on how to use RWEverything to tinker with the EC to manually control your fan speeds. Be well aware that a wrong move could bring your system crashing down or worse, cause physical damage to computer components.
This post also helped me while I was looking for the correct offsets for fan speed control in the EC RAM.
Changes made to the EC are reset after a reboot, so you might want to make a batch script and run it at boot to set the fan speed correctly. Here's what I used for my computer.
The fan speed control offsets in the EC RAM are different for every computer model so you might have to experiment a little to find yours. However, do not attempt to make any changes to the EC RAM unless you fully understand what you're doing and the implications of your actions.