The Big Sky Conference was founded in 1963 with 6 members ... Idaho, Idaho State, Montana, Montana State, Gonzaga, and Weber State.
Colgate plays in the Patriot League for all sports but hockey, which it plays in the ECAC Hockey conference.
Rutgers plays in the Big East conference
The Scenic West Athletic Conference sounds like a powerhouse...
St. John plays basketball in the Big East conference.
The University of Southern Mississippi plays in Conference USA.
He plays in the NFC
Baylor plays in the Big 12.
Sun Belt conference champions and a team from Conference USA.
Indiana State plays division 1-AA football in the Gateway Football Conference.
Both are Division 1A schools. Virginia Tech plays in the Atlantic Coast Conference and Ohio State plays in the Big Ten Conference.
They could, yes... but they don't. They play both sports as part of the Southeastern Conference (SEC). Other schools do play each sport in a different conference. For example, Villanova plays basketball in the Big East, but plays football in the Colonial conference.