As someone who went to Princeton on financial aid, I'll answer this as best as I can.
Princeton, Harvard, and other highly selective universities will give generous financial aid--in some cases, 100 percent coverage of tuition, room, and board--to students who come from lower- to middle-class backgrounds, especially if those students come from rural areas (such as Louisiana) that are traditionally underrepresented at their universities. The student, of course, must have the coursework, grades, extracurriculars, community service, etc. to convince admissions officers to accept that student into their university.
We have a ton of qualified students in and around Acadiana and the state of Louisiana who would get into an Ivy or any of the prestigious non-Ivy institutions AND receive generous financial aid packages because they come from low- to middle-income backgrounds. However, those students typically don't have the support system at home or within their high schools to push them to apply.