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If one doesn't count the 1994 season (which didn't have playoffs or a World Series), the longest streak in MLB history is 14 seasons by the Atlanta Braves (1991-1993, 1995-2005). If one does count the 1994 season, the longest streak is 13 seasons by the New York Yankees (1995-2007).
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Most consecutive playoff appearances
29 Boston Bruins (1968-96)
28 Chicago Blackhawks (1970-97)
24 Montreal Canadiens (1971-94)
23 St. Louis Blues (1980-2002)
21 Montreal Canadiens (1949-69)
20 Detroit Red Wings (1991-2011)
Glascow Rangers (soccer) with 53 championships then Real Madrid (31), then New York Yankees (27)
The New York Giants hold the NFL record of most playoff appearances with 31.
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