The Brookings has a statistical chart which shows in graphic form the impact of family income on children's performance in college. It turns out to be not so important for those with sub-3.0 high school GPAs or above 3.5 GPAs. Few in the former group and most in the latter win baccalaureate degrees, regardless of family income.
Family income seems to matter most for students with high school GPAs in the 3.0-3.5 range. Those from more affluent families do much better than those from poorer families.
These are kids who can go either way, depending on their degree of emotional and financial support. Coaching by parents who understand the higher education process doesn't hurt, either. Hat tip to RealClearPolicy for the link.