I think it's possible to have more than one rooting interest. UT football is far and away the team that I follow most passionately; however, I am also a fan of UNC athletics.
I grew up just outside of Charlotte in the mid-70s and 80s. So I was a Tar Heel basketball fan and that was pretty much it (the running joke around those parts was that the convenience stores were going to be renamed from 7-11 to 0-11 to match the Heels football record!).
In 1990, as I was going into 9th grade, my family moved to Chattanooga. For a couple years I didn't really pay attention to "local" sports. But when UT started recruiting Peyton I was following it because I had some friends that were definitely going to UT after we graduated in 1994. Once that 1994 season started, I paid attention to the Tennessee games to watch Peyton. So, that's when I became a fan, and it didn't hurt that my first 10 years as a UT fan coincided with arguably the best 10-year run in the history of UT football.
I definitely consider myself a true Tennessee fan, and the fact that I've lived and died (most died) with UT football these past 12 years, I think, proves it pretty conclusively.
I'm a fan of athletics for both schools (not a UT football fan and a UNC basketball fan). They've only played once in football, and I was definitely rooting for UT to win the game. In basketball, they've met a few times. I was actually hoping Tennessee would win the game this past December, because I felt like it would mean more to the UT program.
If anyone thinks that makes me "not a true fan", I have approximately zero f's to give about it.