Can you suddenly get an allergy to cats when you never had one before? is it possible?

Yes: Allergic people are not born with allergies. Rather, they are born with the tendency to develop them. Allergies are then developed through exposure to allergens. Why people develop allergies when they do is unclear, but everyone with cat allergy, or ragweed allergy or peanut allergy for that matter, has had their initial reaction at some point.