I love horses and so it is quite hard for me to think of reasons for not getting one, but trying to be objective I will do my best.
1. It costs quite a lot of money to properly look after a horse. The rental of the stables is high if you don't have your own, its food, vet bills, its insurance and all its tack.
2. You really have to dedicate quite a lot of time to a horse yourself and that means exercising it every day or at least visiting it every day to make sure that it is okay, that it has water and that he has been fed. A colic gone un-noticed can kill your horse in very little time so that's why you have to keep a check on him.
3. If you like your social life you will find that you would have to divide the time that you usually spend with your friends with your horse and so hence less social life.
4. It is an extra worry, an extra responsibility and with all the things that you have to do each day anyway you now have another duty - to look after your horse.
5. And last of all if you get one, and one supposes that if you get one it's because you like the animals, you will no doubt get attached to it and suffer every time it gets ill or injured and most of all the day that he is no longer with you as they don't live as long as we humans do.