Apparently, this film is set in a universe where There Are No Community Colleges.
Worse, one of Bartleby's rejection letters comes from Ohio State, which never outright rejects Ohio residents, but gives them a status called 'Options for Ohio Residents', which allows you to enroll in one of the regional campuses and try to transfer into Main Campus later.
And no conservatories, or even any music programs in general. The guy on the accreditation board had to give up trombone for college.
Except his exact words were that he wanted to be a Jazz musician, with the implication being that he had to set aside that dream for something more "practical" while attending College.
Surely to be an Institute of Technology, you'd need technical/scientific courses available.
In this case, it's clear the name was created purely for the acronym, SHIT, so we can let this one slide.
Even community colleges can and will turn applicants away for a pathetic GPA or valid suspicions that they'll be a particularly severe troublemaker. Issues with fake ID cards and the police sound plausible enough.
Why assume that Community College is automatically a "fallback" if you don't get into your choice of a "real" college? Some kids see university as their only option and don't expect they'll fail.
I know it's kind of necessary for the structure of the plot, but it doesn't make any sense that another institution would be allowed to contest an institutions accreditation. Also, non-accredited colleges exist and aren't illegal. Unless they claimed they were accredited, that alone wouldn't make it fraud.