We hear this a lot about the 2004 ALCS - the Yankees ran out of pitching, and all they had left was Kevin Brown.
What is sometimes overlooked is that the Sox starter for Game 7, D-Lowe, had a 5.42 ERA that year (compared to 4.09 for Brown) and was originally left out of the postseason rotation. We were desperate too. D-Lowe came up huge when least expected.