Greatness is a relative term. I think a person can achieve greatness no matter where they are coming from. Making the most out of the cards you were dealt with, and striving to reach your full potential, IMO, makes you a great person. This is true no matter what your background.
I would say, though, on the one hand, that a self-made person is more likely to achieve their personal potential than a person “born” into it, because they already have proven to themselves how much they can achieve, and may be more used to rising to the challenge. A person living the “right” way, only because he was born into it, and raised that way, may be a wonderful person, but has possibly done nothing to reach his full potential. Of course, it is entirely possible that he has grown a lot, in other areas, such as emotional stability, self-confidence, faith, etc., and therefore has proven to himself that he has the ability to reach his full potential.
Everyone has an unbelievable amount of potential. Even the most perfect person, who seems to have no flaw, can reach higher levels. Also, no two challenges are the same, and the same challenge for two different people are not the same, either. Therefore, a self-made person, who has gone through every possible trauma and pain that society has to offer, and has come back to the Truth, may have done the same amount of work as a person born into the most “chashuv” family, never did anything wrong, never went through anything traumatic, but is a shy person, and decided to join a community service that helps the poor. Yes, it seems impossible that the two can even be compared. Of course it seems that the self-made person did so much more! But in reality, we can’t possibly know that. A very shy person’s joining a community service can be just as challenging as another person’s going through a traumatic event. It is our personal associations that cause us to jump to conclusions about the severity level of different challenges.