I don't think height is as much an issue as how experienced a rider you are. If you are just beginning owning a horse and riding I would look more for a horse that is a solid dependable mount with lots of training experience and a good temperament before worrying about how big the horse is.
BSJA have rules about the size of the pony in relation to your age: According to them you should be riding a 13.2 or 14.2 hh.
Which is totally unrealistic as you are rather tall for your age.
But if you are a Happy Hacker then you should consider the suitability more than the size:
Is it bombproof when facing a tractor and trailor on a narrow lane?
What does it do if you are out riding and meet the Hunt?
Etc etc etc
The Pony Club aren't quite as pricky as the BSJA.
I would really advise you to make sure you can handle any horse/pony you try.
Remember that safety is more important than anything else.
BTW I am 50+, 5'2"" and have a 14.2hh heavyweight Irish cob. We did Elementary dressage until she hurt herself and was unable to carry on competing. If I had wanted to di BSJA with her, she would have struggled to make the distances as she would still have had to compete against the bigger, longer-striding horses.
You won't know until you ride a few and decide which one has the best "feel." It's hard to explain. I've ridden ponies and draft horses and no two are ever the same. My favorite was a big App gelding I had in high school.
I think it all depends on the horse. I'm 5'1 and my horse if 16 hands and I love him. My 11 yr old daughter has two horses one is 13.3 the other is 14.1 love them both. She also rides my 16 hand gelding and get's along great with him. Depends on you ability and how much training the horse has, also what you plan on doing with it.
I agree it really depends on your horse and the strength of the rider. My daughter rode a 16 hand Arabian mare but she she broke to the hilt so my daughter only had to move just right to get the desired movement from the horse. I do miss that training level of that horse! I guess I should say my daughter was way younger than that she is and now rides her Dad's Arab stallion Probask(he is a treasure best boy every takes care of his riders). So depending on where she is going to show or ride and what she wants to accomplish
I agree with alot of the people on here. It all depends on the horse. I currently have a 15 hand paint QH that my wife couldnt ride(mare doesnt like women riding on her) and I just picked her up a white with blue spot QH gelding that about 15 hands also....but outweights my horse by about 200lbs. I gotta get mine back in shape and fed up more and shes gotta put hers on a diet. BTW she is 4'11 and im 6'0'' and I weigh about 120lbs more than her.