Since 2001, not counting current seasons:
Red Sox
2003: 2 playoff series
2004: 3 playoff series, 1 championship
2005: 1 playoff series
2007: 3 playoff series, 1 championship
2008: 2 playoff series
2009: 1 playoff series
Patriots
2001: 3 playoff games, 1 bye, 1 championship
2003: 3 playoff games, 1 bye, 1 championship
2004: 3 playoff games, 1 bye, 1 championship
2005: 2 playoff games
2006: 3 playoff games
2007: 3 playoff games, 1 bye
2009: 1 playoff game
2010: 1 playoff game, 1 bye
Celtics
2001: 3 playoff series
2002: 2 playoff series
2003: 1 playoff series
2004: 1 playoff series
2007: 4 playoff series, 1 championship
2008: 2 playoff series
2009: 4 playoff series
2010: 2 playoff series
Bruins
2001: 1 playoff series
2002: 1 playoff series
2003: 1 playoff series
2007: 1 playoff series
2008: 2 playoff series
2009: 2 playoff series
2010: 4 playoff series, 1 championship
In total, that's 67 playoff rounds (including byes) and 7 championships.
I can't imagine that any other city is even close over the same time period.