A third point can come into play in perspective, but only when dealing with extreme heights or lows. Tall buildings are one example. In the case of looking up at a tall building (worm's eye view) the edges of the building will not only recede to the two vanishing points (if looking at corner), but there will be an upward (or downward) recession to a vanishing point. This vanishing point is always directly in front of the viewer at a 90 degree angle to the horizon line. If looking down at an object in three point perspective it is referred to as a bird's eye view.
M.C. Escher uses three-point perspective in the piece Ascending and Descending. In this picture, all the convergence lines (see Linear Perspective: Introduction in Art Studio Chalkboard) recede to vanishing points at the left, right and also below. Even though this is similar to a two-point perspective exterior, he is using the third vanishing point to allow the viewer to perceive the vertical depth. You will notice that all 90 degree angles recede to one of these three vanishing points. This bird's eye view allows one to see planes receding back in three directions.