As of right now, we aren't really sure. Cats and dogs can certainly experience dementia and severe cognitive decline, but not much research has been done on it in horses.
It used to be that most horses didn't make it past 25, but it's now fairly easy to keep a horse alive into his or her thirties, even forties (ponies live longer).
Older horses can show symptoms that are similar to those displayed by, say, dogs with dementia - confusion, unusual dependency on companions, aimless wandering, depression, grumpiness, and even aggression. (You really don't want that in a 1000 pound plus animal, trust me). Horses may also fail to recognize familiar handlers or respond in a familiar way, refuse to be lead or change their eating habits.
(Failing to recognize a familiar handler, however, is equally likely to be a symptom of vision or hearing loss, both of which are common in older horses).
So, the answer is "Probably yes, but we haven't done a lot of research yet."