Well, yes ... and no (to the first part). The sort of extreme-aid programs you mention are called "zero-loan pledges", and they showed up in the mid-1990's as a wonderful ideal. About 70 institutions signed initially, but then the crash happened, and a lot of endowments took a huge hit. Predictably, many of the colleges dropped the zero-loan policy; the most recent number I heard said that only about 2 dozen still offer it. However, even at those institutions, there has always been a catch: you can only get the aid if you are admitted in the first place, and at those schools, it is really very difficult for poor kids to get admitted. The idea of "need-blind" admissions decisions is a very pretty one, but at the 40 or so US schools that are at the top of the heap, it just isn't done that way. Admissions officers can Google just as well as the rest of us, and when your Dad's name pops up all over the WSJ website, you are a lot more likely to get admitted than a student with exactly the same stats whose father's only mention on the web is a lien against him for child support. The major private schools (and increasingly, public flagships as well) do this for reasons that go beyond a particular applicant's ability to pay full tuition -- they want to create legacies; families who attend for generations and whose names end up on buildings. THOSE people are their bread and butter, and they know it. So, the odds that your imaginary student would ever actually get to make a choice between Bama and Harvard are darned small. I can't find the place where I read it, but I heard recently that the number of accepted low-income freshmen eligible for the no-loan deals at Harvard normally is less than 2% per year, out of a class with about 1600 openings. I'm not swearing that's accurate, because I can't cite the source, but from my experience with the internal workings of universities, it doesn't sound far-fetched.
Meanwhile, back at the bank, there are an enormous number of private schools with tuition over $40K, and many of them are not places that are household names. Chapman (in Orange Co., California) comes quite close, but among the schools I can think of right of the top of my head that are well-known in academic circles but not necessarily places almost any American would recognize, I'd include Landmark, GWU, and Wesleyan as examples.
While I certainly would agree that a well-heeled private university is more likely that a smaller school or a public one to be able to bridge an unforeseen financial gap, there are gaps and then there are gaps, and the OP's gap is more on the order of a chasm. Coming up with $4K or so at the last minute is usually doable at a school like that, but coming up with nearly $20K is unfortunately a lot less likely.
