This has been possible since 1947, when the NBA concluded its 1st season. However, the NHL only had 6 teams at the time, and this would remain true until 1967. And Montreal has never had an NBA team, Toronto didn't have one from the end of the 1946-47 season until 1995, Detroit didn't have one from 1947 to 1957, and Chicago didn't have one from 1949 to 1961, and again from 1963 to 1966.
So, for most of the NBA's 1st 20 years, if this feat was going to be pulled off, it was going to be either New York or Boston that did it. And New York didn't do it. They did it in back-to-back years: The Rangers made the Finals in 1950, while the Knicks did so in '51, '52 and '53. But not the same year, at least not for a while.
1. 1957, Boston: The Celtics won, but the Bruins lost.
2. 1958, Boston: The Celtics won, but the Bruins lost.
3. 1972, New York: The Knicks and the Rangers both lost.
4. 1974, Boston: The Celtics won, but the Bruins lost.
5. 1980, Philadelphia: The 76ers and the Flyers both lost.
6. 1991, Los Angeles: The Lakers and the Kings both lost.
7. 1992, Chicago: The Bulls won, but the Blackhawks lost.
8. 1994, New York: The Rangers won, but the Knicks lost in Game 7. This is the closest any city has come to winning both.
9. 2003, New York -- actually, New Jersey: The Devils won, but the Nets lost.
10. 2016, San Francisco Bay Area: The Golden State Warriors and the San Jose Sharks both lost.
11. 2023, Miami/South Florida: The Miami Heat and the Florida Panthers have each made their sports' Finals.
So it's happened 11 times, in 7 different metropolitan areas: Boston, New York, Philadelphia, Los Angeles, Chicago, San Francisco, and now Miami.
Detroit: Possible since 1958, but has never happened.
Washington: Possible since 1975, but has never happened.
Dallas: Possible since 1993, but has never happened.
Toronto: Possible since 1995, but has never happened.
Denver: Possible 1977-82, and again since 1996, but has never happened.
Phoenix: Possible since 1997, but has never happened.
Minnesota: Possible 1990-93, and again since 2000, but has never happened.
Cleveland: Only possible in 1977 and 1978, and didn't happen.
Buffalo: Possible 1971-78, but didn't happen.
Atlanta: Possible 1973-80, and again 2000-11, but never happened.
Vancouver: Possible 1996-2001, but never happened.
Tampa Bay: Possible in 2021, due to COVID restrictions forcing the Toronto Raptors to play all their "home games" outside of Canada, and they chose Tampa, but while the Tampa Bay Lightning won the Stanley Cup, the Raptors did not make the NBA Playoffs.
St. Louis: Only possible in 1968, the Blues' 1st season, before the Hawks moved to Atlanta. It almost happened: The Blues reached the Stanley Cup Finals, but the Hawks were eliminated in the Eastern Division Finals (then the NBA's Semifinal round).
Pittsburgh: Has an NHL team, but hasn't had an NBA team since the league's 1st season, 1946-47. Had an ABA team, 1967-72.
Seattle: Has had an NBA team and an NHL team, but never at the same time.
Carolina: The region could count, but, while the Carolina Hurricanes have won the Stanley Cup, the Charlotte Hornets have never reached the NBA Finals.
Tennessee: The State could count, but, while the Nashville Predators have reached the Stanley Cup Finals, the Memphis Grizzlies have never reached the NBA Finals.
Houston: Has never had an NHL team. Won the WHA title in 1974 and 1975, but the Rockets didn't make their 1st NBA Finals until 1981.
Portland: Has never had an NHL team, or a WHA team. The Portland Rosebuds were the 1st American team to play for the Stanley Cup, as the 1916 Champions of the Pacific Coast Hockey Association, but lost to the Montreal Canadiens.
The following cities have an NBA team, but have never had an NHL team, or a WHA team: Memphis, Milwaukee, New Orleans, Oklahoma City, Orlando, Sacramento, San Antonio and Utah. Indiana has an NBA team, but while it had a WHA team, it's never had an NHL team.
The following cities have an NHL team, but have never had an NBA team, or an ABA team: Calgary, Columbus, Edmonton, Las Vegas, Montreal, Ottawa, Winnipeg.