"Red wine headache" syndrome, as it's often called, affects a surprising number of people. But the culprit isn't sulfites, contrary to popular belief. Sulfite allergies, which are relatively uncommon, can result in wheezing, dizziness, accelerated heartbeat, hives, and, in very rare cases, both of one's arms falling off abruptly, crawling by the fingers across the floor, and trying to throttle one's dog.
OK, the last symptom was a lie. But one thing that is true about sulfites is that your average package of dried apricots contains radically more sulfites than a bottle of wine (about 2000 ppm vs. a legal maximum of 350 ppm; most wines have less). So if you really want to see if you're sulfite-allergic, eat a few dried apricots and check out what happens. Just make sure your dog is parked safely outside.