Most consecutive Conference Semifinals and Conference Finals

[72-10]

Most consecutive Conference Semifinals
22 - Nationals/76ers, 1950-1971
19 - Celtics, 1951-1969
16 - Lakers, 1959-1974
11 - Bulls, 1988-1998
10 - Lakers, 1982-1991
9 - Lakers, 1949-1957
9 - Celtics, 1980-1988
8 - Bucks, 1980-1987
8 - Spurs, 2001-2008

Most consecutive Conference Finals
13 - Celtics, 1957-1969
8 - Lakers, 1982-1989
7 - Lakers, 1949-1955
6 - Lakers, 1968-1973
6 - Pistons, 2003-2008
5 - Lakers, 1959-1963
5 - Celtics, 1972-1976
5 - Celtics, 1984-1988
5 - Pistons, 1987-1991
5 - Bulls, 1989-1993
5 - Warriors, 2015-2019

Of course the league used to have considerably less competition for these top rungs in the league in the 1940's, 50's and 60's before expansion became a norm.

Overlooked in all of this are those early 80's Sixers teams, which in four years made three NBA Finals, and would beat the tar out of the aforementioned mid-70's Havlicek-led Celtics teams.