Ballplayers Who Won Rookie of the Year and MVP

1947 and 1949: Jackie Robinson, 2nd base, Brooklyn Dodgers (1st base in '47, 1st time ROY was awarded)

1951 and 1954: Willie Mays, center field, New York Giants (also 1965 with San Francisco Giants)

1949 and 1956: Don Newcombe, pitcher, Brooklyn Dodgers (also won 1st-ever Cy Young Award in '56)

1956 and 1961: Frank Robinson, right field, Cincinnati Reds (also 1966 with Baltimore Orioles)

1958 and 1967: Orlando Cepeda, 1st base, St. Louis Cardinals (won ROY with Giants)

1959 and 1969: Willie McCovey, 1st base, San Francisco Giants

1968 and 1970: Johnny Bench, catcher Cincinnati Reds (also won MVP in 1972)

1963 and 1973: Pete Rose, left field, Cincinnati Reds (2nd base in '63)

1975: Fred Lynn, center field, Boston Red Sox (1st player ever to win both in the same season)

1970 and 1976: Thurman Munson, catcher, New York Yankees

1967 and 1977: Rod Carew, 1st base, Minnesota Twins (2nd base in '67)

1982 and 1983: Cal Ripken, shortstop, Baltimore Orioles

1977 and 1987: Andre Dawson, right field, Chicago Cubs (won ROY with Montreal Expos)

1986 and 1988: Jose Canseco, right field, Oakland Athletics *

1991 and 1994: Jeff Bagwell, 1st base, Houston Astros

2001: Ichiro Suzuki, right field, Seattle Mariners (won both)

2001 and 2005: Albert Pujols, 1st base, St. Louis Cardinals (also won MVP in 2008 & '09)

2005 and 2006: Ryan Howard, 1st base, Philadelphia Phillies

2006 and 2011: Justin Verlander, pitcher, Detroit Tigers (also won Cy Young in '11)

2007 and 2011: Ryan Braun, left field, Milwaukee Brewers

2010 and 2012: Buster Posey, catcher, San Francisco Giants
(UPDATE since this piece was written in 2012)
2010 and 2014: Mike Trout, center field, Los Angeles Angels (also won MVP in 2016)
2012 and 2015: Bryce Harper, right field, Washington Nationals
2015 and 2016: Kris Byrant, 3rd base, Chicago Cubs

Only Newcombe and Verlander have won all 3 big awards: The Rookie of the Year, the MVP and the Cy Young Award.

Munson is the only Yankee to do it. The award wasn't available until 1947, so Babe Ruth, Lou Gehrig and Joe DiMaggio were never eligible for it.  Mickey Mantle had a troubled rookie season in 1951. Derek Jeter was Rookie of the Year in 1996, but has never been awarded the MVP. He is, however, the only player ever to be named MVP of the All-Star Game and the World Series in the same year, 2000.