As it turns out, it's basically false. I've had an annoying case of contact dermatitis for about two months now, and wound up seeing an allergist to figure out what was causing it. The culprit is a lovely little chemical called para-tertiary-butylphenol-formaldehyde resin. Among other places, it's found in some adhesives that are used for leather and rubber.
Testing for contact dermatitis is done with a patch test. The allergist tapes a whole bunch of tiny samples of common allergens to your back and you keep them on for 72 hours. After that, you go back to the allergist and they remove the tape and look at the reactions. It only took a couple of hours for one particular spot to get itchy, and within a day I was dead certain that the allergen was "that patch right there... the one at the top of my shoulder blade."
Out of 31 patches, that's the only one that came back positive... and it was a really strong positive. That solves a four-decade-old mystery of why some shoes have always made my feet break out. Unfortunatley, I don't have perfect tools to avoid the stuff, since shoes and leather goods don't come with ingredient labels. That's OK, though. I just know that whenever I buy shoes or other leather goods there's about a 10% chance that I'll be allergic. I'm OK with shoe roulette.
I think the last reaction was set off by a deodorant, though. I'm going to have to go home and read some labels.