Never.
The Minnesota North Stars made the finals once in 1990-1991 (Just before moving to Dallas), but were beaten out by the Pittsburgh Penguins in games.
Frankly, the team was lousy (and I do mean bad!) in the regular season (were a dozen or so games below .500) but caught fire in the playoffs. I was a kid during that playoff run, and essentially all hockey fans were cheering on the North Stars. Arguably the last true Cinderella-story the NHL has had.
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Neither the Minnesota North Stars nor the Minnesota Wild ever won the Stanley Cup.
However, after the Minnesota North Stars moved to Dallas in 1993 and became the Stars, they won the Stanley Cup in 1999.
The Minnesota North Stars made a run at the cup in the 1980-81 season, as well as the 1990-91 season. They lost the cup to the Islanders (4-1) and Penguins (4-2) respectively.
The Minnesota North Stars never won the Stanley Cup.
However, after the Minnesota North Stars moved to Dallas in 1993 and became the Stars, they won the Stanley Cup in 1999.
The Minnesota Wild was founded in 1997, but didn't start playing until 2000. The Minnesota Wild have never won the Stanley Cup since it has been a part of the NHL.