Jimmy Stewart and the cat

He was sneezing before he knew the cat was in his office.

It was just an act. I've never known anyone with cat allergies (including myself) who was fine as long as the cat was 3 ft away but started sneezing immediately once the cat was close. Usually the person will start having an allergic reaction when in close proximity or when petting the cat but that reaction is not immediate. It does take a small amount of time for the histamines in your body to begin to attack. I'm not saying it isn't possible just highly unlikely. Had Stewart truly been that allergic to a cat I highly doubt that he would have had so much contact with the cat. And being that he was working with an animal, multiple takes may have been required at which point a true allergy would have him sneezing so often with such watery eyes that it would ruin the shot.
Yes Shep starts to sneeze while the cat is behind him, but that is because the director did his job in directing and getting the timing down.