Starting in the Year 2000, there were five Minnesota professional sports teams playing in the top league in the United States: The Lynx (debuted in 1999), Wild (debuted in 2000), Timberwolves (debuted in 1989), Vikings and Twins (both debuted in 1961).
From 1967 until the Timberwolves debuted, it was just the Vikings, Twins and departed North Stars. Those three franchises made the playoffs in the same season twice: 1969 and 1970.
For a few years before the North Stars bolted for Dallas, there were four major pro teams. The four never reached the playoffs in the same year since the Stars were gone (1993) before the Wolves made the postseason for the first time (1997).
The additions of the Wild and Lynx gave Minneapolis-St. Paul five major teams — four in the long-standing top major men's leagues and one in the WNBA.
Those five teams never have made the playoffs in the same year — defined here as any five total consecutive seasons, since the NBA and NHL span two calendar years. Even with the benefit of that doubt, allowing for bookends on either end from the Wolves or Wild (though both have to be from the same season since I'm making the rules), it never has happened.
The closest it did — and this is legitimately close — was 2003. The calendar year started with the Timberwolves and Wild both making the playoffs. The Wolves lost in the first round to the Lakers, while the Wild made it to the Western Conference finals before bowing out to Anaheim.
That summer, the Lynx made it to the postseason for the first time in franchise history, losing in the conference semifinals. The Twins kept the party going, winning their second consecutive division title. When the Vikings started 6-0, it seemed like a cinch that all five would do it. Then the Vikings went 3-7 in their last 10, missing the playoffs on Nate Poole's miracle fourth-down catch on the final play of a supremely frustrating season.
A year later, four of the five made the playoffs, too: Wolves, Lynx, Twins and Vikings. The Wild missed badly, though, finishing last in its division.