This is the intersection between
need-blind versus need-aware
and
meets 100% financial need vs. doesn't
There are a handful of schools--not many, virtually all of them the creme de la creme of colleges/universities--that are need-blind (make admissions decisions without regard to financial need) and meet 100% of students' financial need (as determined by the college). Some schools are "need-aware"--they do take financial need into account in making admit decisions--but whoever they do let in, they pledge to meet 100% of financial need. Historically, most schools have been need-blind, but they don't pledge to meet 100% of financial need, so you may get in but you might not be able to afford it. The number of these schools, as noted in some of the links below, may be declining though. More and more schools are need-aware and don't pledge to meet 100% of financial need.
Some interesting reading:
http://www.insidehighered.com/news/2013/10/22/george-washington-u-admits-it-incorrectly-told-applicants-it-was-need-blind
http://www.forbes.com/sites/maggiemcgrath/2013/10/29/does-applying-for-financial-aid-hurt-your-college-admissions-chances/
http://www.huffingtonpost.com/kat-cohen/the-truth-about-needaware_b_2758566.html
http://www.thecollegesolution.com/should-you-apply-for-financial-aid-2/
http://www.thecollegesolution.com/need-aware-vs-need-blind-colleges-2/