A laptop fan pulls air through a narrow heatsink. After two or three years it is packed with dust, and the thermal paste between the chip and the heatsink has dried out (sample note).
The laptop then runs the fan at full speed, slows itself down to cool off, and in the end shuts down when you need it most.
We open it, clean the fan and the heatsink, and replace the thermal paste. It takes about an hour and costs from $49 (example prices).