The psi (psi = pounds per square inch, or the maximum air pressure your tire can be inflated to) rating range on the side label of mountain bike tires is just that — a range. Most riders overinflate their tires for off-road mountain bikes out of fear of getting a pinch flat. For most riders a psi setting of 35-40 is fine. However, going too low in psi can lead to side wall cuts much like a tube pinch. So play with psi pressures to find the optimum level that will give you the best traction, control, and protection for the trail you ride on most.
Road bike tires can also be overinflated. Tires are a form of suspension, and while overinflated tires may benefit you with fewer pinch flats, they will slightly reduce your traction, and therefore speed. For example, instead of 120psi, try 100-110. You will gain traction and allow the tire to roll over small imperfections, giving you a smoother ride that is just as fast, if not faster, over the long haul.