An increase in height is possible after the age of 21, but humans stop gaining height once puberty ends, generally between the ages of 16 and 18. However, some people do not finish puberty until their early twenties, states Teen Health Source.
Children grow taller because the body releases a series of growth hormones. These hormones send messages to the cells of the bones and muscles, telling them to multiply and grow. Between birth and the end of puberty, the growth plates on the ends of bones are open, allowing them to respond to these hormonal growth messages. At the end of puberty, the body sends another set of hormones to the growth plates, signalling them to fuse. Once this process is complete, bones cannot grow longer, reports NBC News.com.