I did find one city that had three champions simultaneously: The Detroit Tigers won the 1935 World Series, the Detroit Lions 1935 NFL Championship game (long before the days of the AFL/AFC and the Super Bowl), and the Detroit Red Wings the 1935-36 Stanley Cup. Detroit was still champions in both baseball and football at the time the Red Wings won the Stanley Cup, though technically the Stanley Cup win was in a different season.
Other than that, the closest would be 1969, when the New York Jets won the Super Bowl, the Mets won the 1969 World Series, and the Knicks won the NBA championship in 1969-1970.
And beyond that, there have been just a few teams to even win two championships the same year or season: the 1938 Yankees won the World Series and 1938 Giants won the NFL championship;
The Baltimore Orioles won the 1970 World Series and the Colts the 1971 Super Bowl, which technically was the 1970 NFL season;
The Pittsburgh Steelers won the Super Bowl in 1979 and the Pittsburgh Pirates the World Series that fall;
The Los Angeles Dodgers won the 1981 World Series and were still the reigning MLB champions when the LA Lakers won the NBA championship for the 1981-82 season;
The Lakers won the 1987-88 NBA championship and the Dodgers the World Series that fall;
And the New England Patriots won the 2004 Super Bowl and the Red Sox the 2004 World Series.
Boston has a good chance of pulling off three championships this year: of course the Red Sox are the WS winners, the Patriots should win the Super Bowl, and the Celtics the NBA championship. Don't bet on the Bruins winning the Stanley Cup, though.
Hope you can make sense of that answer!
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