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I saw that it was mentioned once earlier by Robert Smith, but Olin College of Engineering does exactly what you're talking about. The first two classes at Olin received full scholarships for tuition and room (so I only had to pay for textbooks, food, and the laptop), but every class after us receives full scholarships for tuition. The breakdown of the costs for attending are here.
To be fair, we (I'm a student there currently) are much smaller (total undergraduate population of < 300), and were started by the Olin Foundation with the explicit purpose of being free of cost to students while providing a solid engineering background. Whether or not it's feasible to change the very ingrained system at MIT to something like Olin's is questionable. We're only 5 years old, after all.
Needless to say, it is a huge financial burden being lifted off everyone's shoulders - Olin...