For kids from poor or middle class families, Princeton, Harvard, Yale and Caltech will usually cost less than an in-state public school when it comes to what the student and his family actually have to pay.
In-state public school has a lower sticker price, but have less non-loan need-based aid so students have to come up with a high percentage of the sticker price. The top private non-profit schools have higher sticker price, but provide a lot more non-loan need-based aid, so students have to pay a much lower percentage of the cost. At Princeton, for example, the average debt at graduation is only $5k.