I have witnessed some amazing scenerios where a horse by all rights should have panicked remain calm for their rider's sake.
Once I was taking out a trail at a dude ranch and the woman behind me was wheelchair bound, but insisted on riding. She couldn't control her side to side balance, and I noticed her saddle starting to slip. I leaped off my horse and caught the lady just as the saddle was going under her horse. My lead horse didn't take a step, nor did this horse until the lady's husband helped me ease her to the ground, fix the saddle, and remount. Disaster averted, thanks to the horses.
On a sadder note, I also saw a horse mortally wounded by a freak accident on trail. He stepped on a thick branch on the ground which broke and shot up, cutting the major artery on his leg. This horse, aptly named Prince, stood stock still while his rider dismounted. He then walked a few steps, and started running back to the barn. H e made it about 3/4 of the way before he bled out and collapsed.