I fiddled around for a while. Discoveries included the realization that one of the fans I'd recently bought came with four screws that did not fit the fan. The main case fan appeared to be turning quite slowly, though, which seemed cause for concern. I could observe it rotating, maybe only 60RPM instead of 1000's!
The fan was plugged into the power supply on a special "fan only" connection and I guessed that there may be some fan speed logic that had gone bad. Sure enough, when I put the fan onto a normal power supply connection it perked right up. Now, though, I'm worried that the power supply may be reaching the end of its life. Particularly if the power supply's fan is temperature-controlled by the same faulty circuit, though I can feel some airflow coming out of it.