Yep, you can develop an allergy at any point in time.

You might try limiting the cat's indoor territory (If thats possible? With dogs, they say limit them to one or two rooms in the house, preferably a non-carpet area) and thoroughly vacuum and wash other areas. Claritin, or another antihistamine, would help calm your system down so you don't notice the reaction as strongly, if at all.

My dh has a friend who discovered a bunny allergy in his adult life, though he'd had one as a kid. And his allergy is severe...hives, wheezing, the whole bit, on contact...even if the rabbits long gone.