The cold water interferes with proprioception. You cease to directly perceive the arm through that modality and its apparent relationship to you so you are more receptive to other information which contradicts the hypothesis that it isn't paralyzed. When you can trivially directly feel that you can move your arm and that it is doing what you want it to, if you aren't very materialistic you don't question why you want your arm to stay still when you have incentives to move it.