Doctor said husband in allergic to cat...help?

a few years ago my husband had "asthma" for a few weeks....coinsidentally around when we got our cat. The docor said it was asthma becacuse the cat. Well...all of the sudden the symptoms went away and he has been fine for 3 years. so now all of the sudden his symptoms are back onlyi did some research and he has alot of the bronchitis symptoms...His sister has bronchitis at the moment as well...So is it possible its bronchitis ? or does asthma go away and come back years later? because for those years he can pet the cat,kiss her and be fine..i dont get it