NEC has a terrific jazz program, and the double degree program would undoubtedly be a great experience if you could get in. But you can’t. Each year about 20-25 students defy the odds and are admitted to both Harvard and NEC. Of those, about 5-6 are admitted into the joint program. Of those, occasionally (but not always) one is a jazz musician. Not saying you shouldn’t try if you’re at that level. My son does know two jazz musicians in the last six years who were admitted to the Harvard-NEC program. One is there now and one turned them down to go to New School in NYC (which still amazes me). But just understand the odds.
Of course, I’m sure you recognize that all the schools we’re talking about in this thread - Harvard, Yale, Princeton, Columbia, Stanford, Oberlin - are reach or ultra-reach schools, and you should make sure you have some safeties and matches on your list. But Harvard-NEC is beyond ultra-reach (and Columbia has a joint program with Juilliard that I’d put in the same category).