The average height for wainscot is a third of the wall from floor to ceiling. This is usually about 3 feet. This is true for bathrooms, as well as any other rooms in the home.
An alternative height for the wainscot in a bathroom is the same height as the vanity backsplash. Bathroom vanities always have some form of backsplash. It's usually a row of tiles. Carrying the wainscot around the room at this height gives the room a planned, cohesive look.
Although it might seem that installing the wainscot at backsplash height all the way around the room is the best option, it might not be if your tub, such as a whirlpool, is on a platform. You may want to adjust the height of the wainscot to accommodate the tub height.
Similar to the way you can run a wainscot around the bathroom based on the height of the vanity backsplash, you can do the same thing for a wainscot that uses a shower pattern to determine the height, as long as it isn't more than halfway up the wall.