About midway through the Video, atara began complaining that she smelled something burning, and after sniffing around the room, she declared that it smelled the strongest right by my computer. Just as she said that, the sound cut out on the video I was watching. It was pure coincidence, but to be safe, I put down the cat and leaned over to sniff by my computer. There was definitely a burning smell in the area of my tower. It smelled more like burning toast than burning components, but to be safe I decided to shut it down and have a look inside.
Everything seemed in order, though it was a little dusty inside. I last cleaned it about a year-and-a-half ago, so I figured it was probably due again anyway.
I am happy to report that my computer is much cleaner now, runs quieter again (the radiator for the water cooler was plugged up a bit with dust), and everything is working perfectly.
The smell, it turns out, was blowing in from outside. I've been meaning to clean it for awhile now, so somebody's neglect for their toast had a net positive effect.