Texas a&m is currently set to play in their first post season game in program history where they are not eliminated from championship discussions. In their 1939 championship the final AP poll was released after the regular season and that was not changed until 1965. Every other year that it would have mattered they were ineligible for bowls due to NCAA infractions.
Here is a list of around 35 teams that have played a post season game with NC implications before they did.
Regular Bowl games 1965-1991
Michigan State, Arkansas, Alabama, Nebraska, USC, Ohio State, Texas Nebraska, Notre Dame, Penn State, Pittsburg, Clemson, Georgia, Miami, Clemson, BYU, West Virginia, Colorado, Georgia Tech (coaches poll), Washington (coaches poll)
Bowl Coalition (1992–1994)
Florida State
Bowl Alliance (1995–1997)
Florida, Michigan
Bowl Championship Series (1998–2013)
Tennessee, Virginia Tech, LSU, Oregon, Auburn,
Playoff Era (2014-Present)
TCU, Cincinnati, Arizona State, Boise State, SMU, Indiana, Texas Tech, Ole Miss and finally Texas a&m TBD (Tulane Virginia Duke James Madison and North Texas)