I would say that “greatness” (whatever that menas) is usually an attribute of someone who has transcended their upbringing. A truly great person can sit in his room and just look at the walls and come up with a new and powerful new perspective. Or they could be a worldly person who has synthesized many influences.
I don’t like the idea that our upbringing determines whether we’re a good person or bad. We are all responsible for our own actions.
Man was created in the image of Hashem. We are all capable of greatness.