My oldest son is about to turn 20 and is 5'4". Doctors have said that his height should have been 5'9" based upon heredity and other parameters. Beginning at age 5, I became concerned about his height and a couple of other symptoms that were potentially mild endocrine abnormalities. Several doctors (including his grandfather who's a physician) all blew off my concerns.
Well, when he was 15 they started paying attention. By then, it was too late for much increased growth. His growth plates were 99% fused.
He was first wrongly diagnosed with Congenital Adrenal Hyperplasia by an endocrinologist at Cook's. Then, I took him to one of the top specialists in the country in NYC, and that diagnosis was refuted, fortunately. He could have taken human growth hormone at that point, but it would have had some very unpleasant side effects for a boy that age, and would have potentially added 1/2-one more inch. We let him make the decision, and he decided against it. He would have taken it if he could have started earlier and grown more.
I have a couple of things to add to other responses:
(1) Get more than one doctor's opinion. Even with clinical testing, as in my son's case, the doctor can be misinterpreting the results.
(2) Go straight to the best doctor you can find, even if you have to wait a few months to get in. Try to use a pediatric endocrinologist.
(3) Expect to wait a loooonnnnggg time to get in to ANY endocrinologist. I have seen one for blood sugar and other testing. It takes a long time to get an appointment.
(4) There is an adult endocrinologist that I strongly do NOT recommend. I'll give you his name privately, if you wish. He jumped to all sorts of conclusions about my physical condition and symptoms without bothering to ask me about them! This has happened twice, fifteen years apart.
(5) The pediatric endocrinologist at Cook's came highly recommended by a family who got great results for their daughter from her. Again, I'll give you her name privately. However, get a second opinion, because she diagnosed a condition in my son that would have affected his entire life. For one thing, males with that disorder usually can't have children. This would have affected his future marriage, and he might have never known that it wasn't true.
(6) Above all, go with your gut and don't stop investigating things for your son just because a doctor (or two or three or four doctors) blow off your concerns.
I wish you and your son the best.