In the early childhood

I talked to my father yesterday about allergies. He had seen a tv show about this subject, and they said that an allergy starts in the early childhood. If exposed to allergens, like animals, as a child, you won't become allergic.
This can't really be true though, since we had two cats until I was around one year old. My father wasn't allergic either, until around that time. The reason they got rid of the cats was that he developed an allergy towards them. At the age of 27...

How come this is so?