Yes of course you can cut their hair. It all depens upon the country you live in. If the winter times is quite hot, they don't need that much hair, but if it is cold, they need it, or some kind of clothes (what do you call it in English?) So if you want to ride quite hard, so your horse will be sweat, they can be really sick if they have lots of hair, so then I would cut it down. :)
Another thing, if you live in a warm country they don't produce that much hair - they are getting it after the climate, very often.