Yes and no.
Horses don't get spontaneous nosebleeds the way humans can. Bleeding from both nostrils can be a sign of bleeding in the lungs, which happens in some animals after exercise.
But horses can definitely get nosebleeds, where blood (sometimes a lot of it) comes from one nostril. The most common cause is a knock on the head causing damage to blood vessels inside the nose. If a horse has to have a stomach tube, it's not uncommon to damage the nose and cause a nose bleed. These are one off events with an obvious cause and the horse will recover quickly.
If a horse has a nose bleed and is coughing, they may have a foreign body in their nose or windpipe.
Repeated nose bleeds are most often caused by a fungal infection in the guttural pouch - which requires surgery. More rarely, they can be caused by sinusitis (yes, horses can get it), a bleeding lump inside the nose (like a giant blood blister) or, rarest of all, cancer in the respiratory tract.
But most of the time, if a horse has a nose bleed, he probably knocked his head on something like the fence or a tree.