They win the next two games but are not chosen for the playoffs when Georgia beats Alabama and Michigan beats OSU. After beating Oklahoma or another top team in a New Years Day bowl game for a 13-0 record, Alabama beats Clemson in the Championship game making the Irish the only undefeated team in the nation other than Central Fla. Another scenario, other than a loss in one of the last two games, would be a 2012 redux vs Alabama in the playoffs. The Irish are a 20+ point underdog vs Bama.
This scenario is not quite as unlikely as one might think.