I don't think anything is failing, if it was, you would probably be getting BSOD's (Blue Screens of Death) or more severe issues.
You could look around at the motherboard capacitors, and see if any look like they are bulging and/or leaking electrolytes (it would be very noticeable though, and there will probably be some that are bent a little which is normal). However, I doubt they are, because like I said above, you would probably be having worse problems.
The slowing could be a heat issue, like you said, those cases aren't very well-ventilated. Also, once you're doing more demanding tasks like watching videos, the processor will heat up more. I'm not sure if it will work with a retail computer like yours, but I use coretemp to check my processor temperatures:http://www.alcpu.com/CoreTemp/
If I were you, I would just back up my important data, and if signs get worse, then start worrying.
"I think there is a world market for maybe five computers." - Thomas Watson, Chairman of IBM, 1943