Bones are arguably the strongest part of the human makeup, but how it "grows" is dependent on different situations.
Puberty - I assume you are talking about this one, the bones have "growth plates," cartilage in the ends of each bone. When we are born, the baby's skeleton is practically cartilage, but as time goes on, it hardens and lengthens in conjunction to pubertal phases. Lengthwise, the pituitary gland releases HGH (Human Growth Hormone) into growth plates, causing them to expand. Human Growth Hormone can be prescribed to children of a short stature/hormone deficiency, but it's only before puberty, before growth plate closure, or being hormone deficient. However, the side effects outweigh the benefits in healthy individuals, and would definitely cost more in total in money+time than one lengthening procedure. The growth plates can be checked with an X-ray of either your hands or your knees. If the X-ray shows no sign of growth plate, then you're most likely done, your only choice is either acceptance or surgery. Or, if you are not willing to pay for an X-ray, see your hair patterns. Did thicker hair grow in your forearms, legs/thighs, face, privates, chest, fully developed privates? Is there a snail trail? Check out the tanner stages scale online, each stage has different growth rates, 5 being the slowest. While these studies for the scales have been accurate, some lucky ones have insane growth spurts even when genitals are fully developed/hair patterns and physiques of adults. Your genetics determine your rate of growth, when puberty starts/finishes, and what height you will become. Growth can never be enhanced, but only stunted. If you are still growing, try to avoid stunting it as much as humanly possible (Smoking cigarettes/lack of sleep are the biggest stunts)
Broken legs (Surgery) - Bone broken? No problem, bone grows and regenerates what has been broken if not even stronger. Using this to doctor's advantages, we are able to use internal/external fixators on either the femur (thigh) or tibia (lower leg).. Bones are "Broken," and are monitored through devices that will expand the broken gap. Over time, the broken gap is surrounded by bone and is lengthened as a result. Pain and discomfort is at high, but keep in mind the legs are practically broken in half the whole time. It's not 100% for some doctors, the bone can grow in unwanted places, but good doctors can lower that risk.