Bad news in Playstation-land. Rhing, I appreciated the link you passed along above, but the process to tune the player was beyond my technical abilities. So I bought a second Playstation on eBay. And it turns out that not only would it not play burned CDs either as I described above, but after a few days it started skipping the same way (though not as badly) on perfectly good new CDs. So I put the old one back in the system, and after a day it started doing the same thing. So I'm thinking the inability to play burned discs may have been a symptom rather than the problem itself. I'm bummed.
In any case, though I love the sound of the Playstation I'm looking now at spending a few bucks more on something a little more reliable and less quirky in its operation (and with a remote! and a non-plastic case!). I'm thinking of an Onkyo DX-7555, though I'm going to take some time looking around. I'll post a comparison when I'm able.