Yes, horses can get blood clots. Anything that has blood in its body can clot its blood. Blood clotting helps heal your body, but some times too much clotting or clotting at the wrong time happens. This is when clotting is dangerous. Horses, people, monkeys, and birds all have blood, so they can clot. However, some living organisms do not have blood like you and me. Insects for example do not have blood!