Yes, it is possible to test positive to an allergen through both skin and or blood tests and not be allergic. It is also possible to test negative and actually be allergic.
The only way to tell if an allergy is a true allergy is if there is a reaction history to the allergen in question.
Sometimes individuals are tested for multiple allergens and some of those allergens show up as positive on allergy tests even though the individual is consuming them fine. This usually means the individual is not allergic to that allergen.
However in the instances of no reaction history and no exposure history–it becomes difficult to ascertain whether the individual has a true allergy or not.
For some individuals, an allergist may recommend a food challenge to determine whether there is a true allergy in the absence of reaction history or if the individual accidentally ingests the allergen in question without reaction.
If you think you might not be allergic to a food you have tested positive for you should discuss this with your allergist before attempting to consume the food/allergen.