Yes, older laptop fans can get clogged with dust bunnies, cat hair, etc. and slow down, struggle or even stop, which then causes overheating. Overheating can be caused by other hardware like heat sink faults. Sometimes you can overcome this by directing a fan straight onto the hot side of the computer to cool it down.
I usually set my laptop on a book smaller than the laptop, so there’s about an inch of space between laptop and table, all around the edge, and the fan vents are unblocked.
When I forget and the computer gets hot and starts panting, the quickest way to cool it off is to hold it upside down for a minute.