I was asked the following question on Quora.com earlier today:
How many times has a Cy Young award winner gone 0-2 or worse in the post-season?
My response was as follows:
Well, it’s happened to the best. Obviously, I am assuming this question was posted following Los Angeles Dodger lefty Clayton Kershaw‘s second loss in the 2014 NL Division Series against the St. Louis Cardinals, a defeat that sent L.A. home for the winter.
In 1997, Seattle’s Randy Johnson, the AL Cy Young winner two years earlier (and a 20-game winner in that current season), went 0-2 against Baltimore in the Division Series as the Mariners lost three games to one. The following year, Johnson was 0-2 for Houston in the NL Division Series versus Kevin Brown’s San Diego Padres, with the 102-win Astros embarrassed in four games. (In 1997, Johnson was 20-4 with a 2.28 ERA, finishing second in the Cy Young race to Roger Clemens. In 1998, the Big Unit was 10-1 with a 1.28 ERA following a late-season trade to the Astros.)
Speaking of Brown, his wild-card Florida Marlins beat Greg Maddux twice in the 1997 NLCS, giving the four-time Cy Young-winning Maddux an 0-2 record in that series. The Braves, who won 101 games during the regular season and finished nine games ahead of Florida in the standings, lost to the Marlins four games to two. Maddux had won the Cy Young in 1992, 1993, 1994, and 1995, and was 19-4 with a 2.20 ERA in 1997. 
Brown’s 1998 Padres also handed the Braves’ Tom Glavine an 0-2 mark in the NLCS, with San Diego knocking off Atlanta in six games. Glavine had won the Cy Young in 1991 and would win it again that same 1998 season with a 20-6 record and a 2.47 ERA. The Padres had a good year with 98 victories, but they were underdogs against the Astros (102-60) and the Braves (106-56). San Diego’s brilliant run ended in the World Series, where the Padres were swept by the 114-win Yankees.
In 1993, the Toronto Blue Jays beat Cy Young winner Jack McDowell (22-10, 3.37 ERA in the regular season) twice in the ALCS as his Chicago White Sox went down four games to two. It was McDowell’s second straight 20-win season that year, and he was named the 1993 Cy Young in the offseason. McDowell was 0-2 with a 10.00 ERA in that 1993 ALCS, and then two years later in the AL Division Series with the Yankees against the Mariners, was 0-2 with a 9.00 ERA (with the second loss coming in relief in the decisive fifth game).
So, it’s happened before. I’m sure others will chime in as far as exactly how many times it’s happened.
 1997 was truly an odd season. In addition to Maddux and Johnson, a couple of other top pitchers went 0-2 in that year’s postseason. Brown himself went 0-2 in the World Series against the Cleveland Indians. Though Brown never won the Cy Young, he was runner-up to Atlanta’s John Smoltz a year earlier, and was also a 21-game winner in 1992. Andy Pettitte of the Yankees, meanwhile, was 0-2 also in the AL Division Series against Cleveland, one year after finishing second to Pat Hentgen for the AL Cy Young. The Indians also won two games in the 1997 ALCS that were started by Baltimore’s Mike Mussina, who got a pair of no-decisions as the Orioles couldn’t score in those two contests.