Allergy to cats??

For the past month or so, my son has been getting a cold every weekend. It'll start sat/sun; symptoms include runny/stuffy nose and coughing. It'll last until wednesday, then he's fine. On the weekends our mil comes over, or we go over to her place and she has a cat. She lives in a basement apt, so the space is limited and this cat is overweight so it can't groom itself efficiently, causing her hairs to get EVERYWHERE. Nothing changes over the weekend with routine, habits, activities that would cause a cold consistently.. that's the only thing I can think of. Has anyone had experiences with pet allergies themselves or with their kids. Could it be that severe? I just feel so bad that he's been on/off miserable this past month. He can't breathe, eat or sleep properly because of how wishywashy his nose gets. :(