it could be overheating since you say it gradually gets worse. get HWmonitor to monitor your cpu and gpu temps. If you have had the PC for a while you may need to clean the dust from all the gpu and cpu fans and heatsinks and also clean out the PSU.
Thanks to everyone for the suggestions.
I opened the case to give everything a good dusting and to look at the video card, and lo and behold, the video card fan isn't spinning and is essentially a heater at this point. I plugged the monitor and power back in and started up a graphics-heavy game at the highest settings and waited for things to start lagging. When they did, I gave the video card a quick spritz with the cold compressed air -- instant fix. As for a long-term fix, I think my only option is a new video card.