Harvard's endowment is $32 billion. They can fund many students at $50,000 a year from the interest without touching the principle, and it's a major goal of theirs to do so. Yale, Princeton, and Columbia also have multi-billion dollar endowments and are aiming to do the same. They can afford it.

This is how the University of Chicago does it (from their website):

Thanks to an extremely generous gift of $100 million from an anonymous alumnus, we are now able to offer an even more competitive financial aid package to many of our students. The Odyssey program has allowed more students from modest backgrounds the ability to pursue all that the University has to offer without the worries of working to pay off loan debt. For students with family incomes below $75,000 a year, all expected federal student loans are replaced with Odyssey Scholarships. For students with family incomes between $75,000 and $90,000, half of their expected federal student loan obligation is replaced with Odyssey Scholarships. You must apply for need based financial aid in order to be considered for this need-based University funded scholarship.