Is it possible that a child could overcome an allergy completely, over a few years?

Sometimes: Some kids outgrow food allergies overtime. It is not uncommon.

Define overcome !: Some kids with simple nasal allergies become less symptomatic when older (bigger nasal passages) and able to blow their nose well & avoid things that trigger them. Some foods can be less of a problem and some not. Things like a peanut allergy can be life threatening and life long.Simple blood tests can identify which foods are dangerous and which less so.Your doc can sort this out for you.