Okay, so I got my old violin back out. For kicks, I decided to see how close I could tune it to what it's really supposed to be. Picked something that sounded reasonable, and started going. Kept feeling it was a bit flat, but I wasn't sure, and I didn't want to tune it too tight, since it's been detuned for at least 2 years. Especially when I realized partway through that the bridge was almost a centimeter off-center. Oops. Slid the bridge back, and kept going.
When I was done, I still felt it was too low, but couldn't find anything that felt better by going up a note or two, so left the "a" as-is. Got on the 'net and found a 440 Hz mp3. Yeah, it was several notes higher than my "a", so I fixed the a-string. Then decided to see how far it was off, by sliding my finger up from 4th position on the d-string (4th on the d produces the tuning a, if the d-string is correct). Kept going until precisely when my finger hit the only tape marker left on the violin (I pulled the others off when one got sticky years ago and I thought I was ready to give up that learning crutch. Wish I had 'em now).
Wacky. I didn't manage to hit the 440 Hz a, but I'm pretty sure that's a 7th position tape, as it's just shy of the octave-harmonic spot. So I managed to tune the darn thing exactly a third low, and knew it was low, though couldn't figure out exactly how low, and going up "a note or two" made it worse.
Okay, I'm slightly freaked out now. Seems the hours I spent trying to learn to have decent pitch actually had some effect. I still can't sing a pitch worth a damn -- heck, I couldn't even listen to the 440 and then sing it. But I could play it.