(prior to realignment) had a similarly stair-stepped bracket. The top 4 teams had byes into the quarters while the lower seeds could've played 2 previous games at that point.
For example, see the 2010 bracket:
WikipediaThe 2010 Big East Men's Basketball Tournament, a part of the 2009-10 NCAA Division I men's basketball season, took place in March 2010 at Madison Square Garden in New York City. The West Virginia Mountaineers defeated the Georgetown Hoyas 60–58 in the tournament finals to receive the Big East Conference's automatic bid to the 2010 NCAA Tournament. It was West Virginia's first Big East Tournament championship. This was the second Big East tournament to include all 16 of the conference's teams. The teams finishing 9 through 16 in the regular season standings played first round games, while teams 5 through 8 received byes to the second round. The top 4 teams during the regular season received double-byes to the quarterfinals.
https://en.wikipedia.org/wiki/2010_Big_East_Men%27s_Basketball_Tournament[https://en.m.wikipedia.org/wiki/2010_Big_East_Men%27s_Basketball_Tournament]
The Big East is a little more balanced now that they have a smaller conference, but I'm sure if you looked around you'd find plenty of other conferences with strange setups.
As others have said, it's mostly to protect/reward the top seeds and set them up as well as possible for the NCAA tourney.