About 60 colleges claim to meet 100% of demonstrated financial need.
(http://www.usnews.com/education/best-colleges/paying-for-college/articles/2014/09/15/colleges-and-universities-that-claim-to-meet-full-financial-need)
Almost all of them are very selective private schools.
Do a little arithmetic to estimate the number of students admitted each year to these ~60 schools.
Brown accepted ~2700 students in 2014.
Colgate accepted ~2300.
So there must be over 120K offers of admission, collectively, from the ~60 “full need” schools (with many overlapping cross-admits, of course). The number of freshman places might be about half that number (~60K enrolled students).
Among college-bound seniors in 2014, approximately 76,000 students scored 700 or above on the SAT-CR.
(https://secure-media.collegeboard.org/digitalServices/pdf/sat/sat-percentile-ranks-crit-reading-2014.pdf)
If we use 76K as a crude proxy for the number of “top” students graduating from US high schools each year, and 60K as a rough estimate for the number of places available in “top” colleges … then it appears that there is a place in some “top” school for most of the “top” students who want one. Not nearly all 76K high-scoring seniors even want (or could afford) one of these places. Plus, there are many more places at colleges that are nearly as selective as these ~60, with financial aid nearly as generous (and many more offers than there are places). All these schools dip well below the ~700 SAT-CR mark to fill their seats; apparently it is not just to give away places to “hooked” applicants that otherwise would go to high-stats applicants. There simply aren’t enough available-and-willing high-stats applicants, apparently, to fill all the places at all these colleges.
Nevertheless, to be on the safe side, most applicants ought to play their cards across a wide selectivity range within this ~60-college space (with safeties chosen from among affordable, less-selective schools outside this space.)