Update: Well after a second drive started having startup problems, I started looking for a common problem that was not the disk drives. I swapped out the IDE cable, poked the motherboard connectors, then finally measured the power supply voltages. Whoa. My supply was measuring only 4.5V on the 5V line at cold start. After a couple of minutes, it drifted up to 4.7V. I've never seen that before. But after swapping out the power supply, an Antec Truepower that came with the Sonata case that I have, everything worked fine.
The Seagate disk was already on the way back to Seagate when I figured out the problem. So what does Seagate do when they discover nothing wrong with a returned disk? Will they send it back?