We know that TCU can't win the BCS National Championship because the Horned Frogs didn't make the BCS National Championship Game. That's unfortunate, and reason for folks in Fort Worth to think up signs like Another BCS Catastrophe to display at the Rose Bowl Game should their team crush Wisconsin in Pasadena.
And while that doesn't technically mean TCU can't win a "national championship," it does make those chances slim and preposterous.
The chances that TCU wins a national title rely almost entirely on the AP Poll. USA Today coaches poll voters are obligated to vote for the BCS National Championship Game winner, and Auburn and Oregon are 1-2 in literally every ratings system used in the BCS. TCU isn't jumping those schools in those ratings, even if it gets a huge win over Wisconsin to go with a lackluster Auburn-Oregon tilt.
So TCU must hope that its win impresses the AP Poll voters like USC did at the end of the 2003 season; it's the last year in which a AP national championship was awarded to a team that didn't win the BCS National Championship. The problem with that: those Trojans only beat Michigan 28-14 in that year's Rose Bowl, and won their title because they were great against good competition all year, and arguably better than two one-loss teams at the end.
The Trojans beat three top-six teams in Auburn, Washington State and Michigan, and moved past two one-loss teams — BCS National Championship Game participants Oklahoma and LSU — to take pole position in the AP Poll before the bowl games, because of Oklahoma's stunning loss in that year's Big 12 Championship Game.
TCU's situation isn't analogous: the Horned Frogs' best win is over a Utah team that lost to Notre Dame, and no matter who wins the BCS National Championship Game, they will have to jump an unbeaten team in the AP Poll to pull off the split national title. The better comparisons for 2010 TCU are 2008 Utah (second with 16 first place votes in the final AP Poll of 2008) and 2006 Boise State (fifth with one first place vote in the final AP Poll of 2006), unbeaten teams from non-BCS automatic qualifier conferences that didn't even finish second to BCS national champions. (Those teams did finish first in some computer rankings, but that's far less likely to happen for TCU.)
And, well, there's also the smell test: Oregon will have beaten Auburn and Stanford if it wins the BCS National Championship Game; Auburn will have beaten Oregon, Arkansas, LSU, Alabama, and South Carolina twice. Those records are better than TCU's would be, even if a motivated Horned Frogs team takes out frustrations on Wisconsin to the tune of a 60-point win. A massive Rose Bowl win plus a close BCS National Championship Game might get TCU a handful of first-place votes in the final AP Poll, but it won't get the school more than either Auburn or Oregon would receive.
Sorry, Horned Frogs. A split national title this year just isn't going to work out. There's always the Big East in 2012, I guess?