Actually there are only about 50 colleges in the country - not including public institutions - who are truly "need blind." The rest are "need aware" which means they aren't looking to fill up their school with full freight students but do want to have a certain percentage of full payers and students with lower financial aid needs in order to balance the books.
Johns Hopkins is pretty open about their "need aware" policy on their financial aid web site --- they say something to the effect that if they have two equally qualified candidates and they've already doled out their limited financial aid funds, they'll give slight preference to the candidate that doesn't need aid. It's not that they don't want a diversified student body, it's just that they don't have unlimited funds.
Macalester is going through a huge debate on just this subject right now - they have been need blind but have run into something of a crunch trying to be both need blind and promise to meet 100% of need of admitted students --- so they are considering becomming "need aware" - but they will still promise to meet need of admitted students. Reading between the lines - they'll still take plenty of financial aid students but will tend to look closely at balancing things by admitting more no need or little need students.
The short answer: being able to pay full freight won't get you in if you don't have the qualifications to begin with, but it MAY slant things in your direction if it comes down between you and a candidate with similiar credentials who needs a lot more money.