There is truth to the idea that difficult experiences can bring out certain traits in ourselves more strongly. But I'm not sure I agree with the statement, "you become more the person you really are". That implies that if you fall apart after a certain event, you were "truly" a weak person to begin with, and if you cope with it admirably and even discover new strengths in yourself then you were "truly" a strong person.
We're all a mixture of strengths and weaknesses, and adversity can bring out either, or both. It depends on a number of factors, such as the amount of support and positive influences in your life, how motivated you are to push through pain or discomfort rather than avoid it, and what else is going on in your life at the time.