Wow, thanks for all the replies, folks!
@dougy3,
Yes, I am indeed using resistors on each string to limit current. And, yeah, I think it's really weird that the string fails after a couple minutes, yet next time I plug it in (well, after waiting several minutes at least), it repeats the behavior, i.e. the led isn't (yet) "blown"!
@SABorn
Okay, the LEDs I'm using specify 3.2V typical, 100ma max, and 5V reverse, described as "1/2 watt".
I have six in each string, powered by a supply from some Altec Lansing computer speakers, "class 2", rated 15VDC @800ma, though without a load it measures 19V. I'm using 150 ohm resistors, which keep the current in a low range of around 15ma, and the LEDs appear perfectly bright.
Again, the other three strings, wired identically, work just fine. I'll probably just disconnect each LED in turn in this string to test separately, probably just powered by a battery.