33, Los Angeles Lakers started its winning streak on November 05, 1971 when they beat the Baltimore Bullets, 110-106 and went to win until January 07, 1972 when they beat the Atlanta Hawks, 134-90 and two days later, Kareem Abdul-Jabbar and the Milwaukee Bucks finally ended the winning spree by scoring a 120-104 win and snapping Lakers' 33-game winning streak.
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The San Antonio Spurs won 12 playoff games in a row in 1999.
In other news:
The best record for an NBA champion belongs to the 2001 LA Lakers, where they went 15-1 (they are also the only team in NBA history to reach the NBA finals undefeated, they lost the first game to the 76ers in the finals but they won 4 games in a row after the lone loss). Looks like I forgot that the '88-'89 Lakers also made it to the finals undefeated, only to lose to Detroit in a sweep. (Magic Johnson and Byron Scott got injured in the conference finals.)
And in case you were wondering the best regular season record is the 1996-97 Chicago Bulls where they went 72-10 and the longest winning streak for an NBA franchise is 33 games in a row (Which includes the longest road winning streak) for the 1971-72 LA Lakers, they also finished the season with a 69-13 record (which stood until the 1996-97 Bulls with a 72-10 record) and won the championship.
15-0 by the Washington Capitols in 1948-49 season (November 3-December 4, 1948)
15-0 by the Houston Rockets in 1993-94 season (November 5-December 2, 1993) a tie
11 games
The 07-08 Houston Rockets with 22 consecutive wins.
In the 2008-2009 season, Boston went on a 19-game winning streak, which is a franchise record, and fifth in NBA's all time winning streak records.
44-Chicago, March 30, 1995-April 4, 1996 (7 games in 1994-95; 37 games in 1995-96)
The Boston Celtics' streak of 8 consecutive NBA championships is the longest streak of consecutive championships in U.S. professional sports history