First of all, I was playing on the latest snapshot: 16w06a.
I was riding my horse over hilly, forested terrain, and tried to jump up a three-block wall. The jump bar worked as it should and the rearing animation played, but the horse didn't actually jump. I suspected I just timed the jump badly, so I tried it a few more times, but to no avail. I suspected that I might have been clipping a tree, so I readjusted my position and tried jumping a couple more times. It didn't work. So, I decided to take a side route, which involved going under leaves. The moment I went underneath the leaves, my horse and I were insta-killed. Curiously, the death message was "mvmlego hit the ground too hard".
MC-96815 Horses do not jump!