Though a handful of teams have.
Oregon State is playing in the CBI this year. In fact, they're looking to defend their 2009 CBI title. Stanford played in it in 2009 with Oregon State, and UVA and Washington both played in the 2008 tourney. In the SEC this year, Alabama would be eligible for the CBI, but I'm sure they would've turned it down if offered. We were eligible last year, and I don't know if they tried to get us a bid, but we definitely would've turned it down.
The CIT is actually designated for mid-majors only, so they don't invite major conference tourney teams. USM is in the CIT though this year. Maybe they could be NITCITChampsburg.