The tedious way would be to go through the Forbes 400 list and attempt to categorize. Eyeballing the top 20, six are clear mega-inheritances (3x Waltons, 2x Kochs, 1x Mars).
For the rest, it depends on where you want to draw the line for "modest backgrounds". For example, which side does Bill Gates fall on? He's certainly more self-made than Christy Walton, in that he inherited only millions, not billions. But having a multi-million trust fund from your bank-founding grandfather, and a mother who's on several boards of directors, is still considerably above the median in terms of winning the birth lottery (http://philip.greenspun.com/bg/).
For the rest, it depends on where you want to draw the line for "modest backgrounds". For example, which side does Bill Gates fall on? He's certainly more self-made than Christy Walton, in that he inherited only millions, not billions. But having a multi-million trust fund from your bank-founding grandfather, and a mother who's on several boards of directors, is still considerably above the median in terms of winning the birth lottery (http://philip.greenspun.com/bg/).