The issue with contamination
Contamination while a product is being created is also highly likely even with your favorite gluten-free brand. Nowadays, there is almost no such thing as 100% gluten-free, but there is definitely an acceptable amount of gluten that a patient can have without a reaction. Avena Sativa (Oat in Latin) is another thing to consider. Although it is typically considered gluten-free, some patients have still reported reactions to it. Thus, it is best to avoid products that are made with oats.
But, could the concerns of celiac patients really be just a product of their overly-cautious minds? Probably not. But note that gluten can trigger several other allergic reactions that are not from celiac disease. It may be that one has developed another sensitivity to gluten resulting in an allergy. The best way to make sure is to consult with your physician or doctor.