Sometimes. While a player will usually feel a slight sting that will quickly fade - similar to a firm flick on the arm - paintballs can also cause welts and bruises depending on the speed of the ball, the distance the ball travels, and where on your body the ball hits. To keep things fun and safe, be sure to keep your gun firing slower than 280 feet per second, and never shoot people from less than 20 feet away.
How much a paintball hurts also depends on how much padding or protection you wear. If you are only dressed in jeans and a t-shirt, expect small bruises that will fade in a few days, while a sweatshirt or other thick clothing will usually prevent bruising. Some people will wear protective vests, but I have generally found these to be unnecessary.
Paintballs that break (or worse, bounce off and don’t break) on bare skin will hurt, but this can be avoided by wearing gloves to protect your hands, backwards baseball caps to cover your neck, and long-sleeve shirts and pants to protect your arms and legs. Your head is already protected by your mask, and a shot to the goggles rarely causes any pain.
Most severe bruising occurs from guns that are shooting hot (too fast) or from shooting from too small a distance. If you are concerned about getting bruised, wear multiple layers and you shouldn't have any problems.