Do horses get stressed?

Absolutely. Horses like routine, and can get stressed when their routine changes (One of the qualities of a good show horse is a high tolerance for stress and being able to deal well with being moved around, being in different stalls, etc).

Stress can also be caused by training - many young or green horses go through a "rebellious" stage at some point during their education, which may be caused by being overwhelmed by training. Most commonly, though, I've seen it when the horse first starts to learn correct carriage - but are unable to physically perform it because the only way to develop the muscle to carry a rider correctly is to do it, and thus the horse becomes frustrated and stressed. Stall rest can cause major stress for a horse. So can separation from a long-term companion. Some horses also become stressed when transported, and there are strong indications that this may be caused by poor hauling conditions (If your horse won't load, you may just be a bad driver).

Some horses get stressed if they are handled and ridden by different people, or forced to deal with people or horses they don't like. The final cause of stress is rider/handler stress - the horse can pick up on this and get stressed themselves.

Symptoms of stress in horses can include weight loss, stall or fence walking and other stable vices, excessive yawning, tooth grinding, bad behavior under saddle, diarrheoa, excessive urination, trembling/shaking, excessive licking or chewing, elevated pulse and respiration, "breaking out into a cold sweat," bolting their food, biting and even ulcers.