The One With The Ball - S5-E21
Corrected entry: While Sphynx cats are not generally hypoallergenic, Monica could not have had an allergic reaction because her exposure to the cat was minimal and she didn't touch it.
The One With The Ball - S5-E21
Corrected entry: When Rachel is holding the cat on the cushion with the red oven mitt, for one shot her hand changes from holding the cat, to holding the pillow. (00:12:00)
Correction: I just watched this episode, and you can actually see her move her hand from the cat to the pillow and back again.
The One With The Ball - S5-E21
Corrected entry: In The One With The Ball, Monica ends up showing she has an allergy to cat dander. But in the previous season, in The One With The Cat, Phoebe finds a cat that she thinks is the reincarnated spirit of her dead mother. Monica is in close contact with the cat, and even petting it in one scene. Now, it's possible to develop allergies over time, but the time between these two events is very short (in storyline time).
Correction: Monica never says that she has an allergy to cats. It's more likely that she sneezed because of the cat dander but doesn't have an allergy.
Correction: Many people, like my mother, are highly allergic to a cat's dander or saliva and will have a reaction just being in a room where the cat had been. No actual contact is needed in order to have a reaction. The cat doesn't even need to be there at the same time as the allergic person. Monica could easily have still had an allergic reaction to the cat eve though her exposure was minimal and she didn't touch it.