Yes, it is possible. Happens all the time, in fact. You hear people say, 'But I have taken X for years and never had a problem,' but there's a good physiological reason for why it does happen:
Immune reactions and allergies--almost without exception--require *prior exposure* to something. IOW, the immune system has to be acquainted with a substance in order to react to it and make antibodies. That's the essential nature of immunity.
The mystery is *why*, after tolerating something like a medication, a food, or even a cosmetic for years, the immune system actually makes a mistake and decides the substance is a bad guy after all.
For more information, check out this site, especially the paragraph titled "What causes allergies to start?":
http://www.webmd.com/allergies/guide/chronic-allergies-causes
I believe the answer is unlikely but possible. One should always be on the outlook for unusual side effect symptoms.
I have several heart drugs that I have taken for years and I am always on guard and on the outlook for signs of symptoms or increases in symptoms problems.
I do not take antibiotic often, but I have never had any side effects or reactions that I could detect.