Totally depends on how they experienced "being a rich girl", you know?
If its some dumb chick whose daddy just hands them cash left and right without questioning it so they can go on shopping sprees or whatever, she is probably going to be a stuck up brat who doesn't understand much about real people in real life.
On the other hand if she grew up with a conception of what money is and had reasonable parents who HAPPENED TO BE RICH, that is far different. Sure, she may buy gray goose instead of Rubinoff, but she'll buy it to share with everyone...that kind of thing.
I know both types, so again, like most stereotypes, it comes down to who you are as a human being and what you've experienced rather than a cut and dry classification.