Usually when we call a horse "crazy" or "loco" we mean it's hard to train.
Can horses be mentally ill?
There's no simple answer to this question. Obviously, we can't ask them how they're feeling and we don't really know what's going on in their heads.
However, I've personally observed horses demonstrate behaviors that pretty closely resemble symptoms of depression, anxiety and even PTSD.
Horses certainly experience the effects of trauma. For example, take the case of Anglo Starsky, a Stakes winning Standardbred gelding who ran for several seasons in the UK. His career ended in an on track wreck which resulted in him flipping over and the sulky landing on top of him.
Starsky never went between the shafts of a wheeled vehicle again. He outright refused to ever allow himself to be driven. (He was retrained as a saddle horse and turned into a quite decent low level hunter). His behavior clearly indicated that he was now afraid of being driven, in much the same way that I'm afraid of wasps after being stung multiple times as a child.
My opinion on the matter is that as best as we can tell, horses can and do suffer from emotional disorders. In some cases these are even treated with psychiatric medication. However, they probably don't experience more complicated issues such as disassociative personality disorder (what used to be called MPD), etc.
We can't really tell, though. We can only tell what they're doing.