They came close.
Twin brothers John and Jim Sevcik were in the team's minor-league system in the 1960s, but only John got called up. And former Twins owner Calvin Griffith had twin brothers (Jimmy and Billy) working in the organization.
Now, identical twins Stan and Stew Cliburn are manager and pitching coach of the Twins' Double-A Eastern League affiliate in New Britain, Conn.
The 44-year-old Cliburns have already caused some confusion in the Rock Cats' locker room.
"We've had our moments where we've messed up, I've called Stan Stew and Stew Stan a few times," said Michael Restovich, an outfielder.
The Cliburns are used to the confusion. The twins played with each other in the minor leagues. Stan was a catcher, Stew a pitcher.
They both made it to the majors, but not together. Stew pitched in parts of five seasons for the California Angels. Stan played in 54 games for California in 1980.
"I've signed many a kid's baseball card with Stew just to not make the kid sad," Stan said.
Rock Cats players said being a twin could have its advantages, especially for a manager.
"What we're waiting for is when Stan gets ejected, then go in and put Stew's uniform on and come back out and manage," said third baseman Michael Cuddyer.
Talk about double trouble.
University of Georgia basketball players Coco and Kelly Miller, twins from Rochester, Minn., were part of "Twins Day" last May at the Metrodome in Minneapolis.
About 500 sets of twins attended the game, getting in for the price of one ticket. Others who could have used the discount:
Basketball players Tom and Dick Van Arsdale, and Horace and Harvey Grant.
Minor-league baseball teammates Jose and Ozzie Canseco.
Pro football players Tiki and Ronde Barber.
Stanford basketball players Jarron and Jason Collins teamed up in February as Players of the Week in the Pacific-10 Conference.
But they're hardly the first twin teammates in college basketball. The long list includes Dwight and Dwayne Praylow at Wichita State; Jon and Joe Ross at Notre Dame; Sammie and Simeon Haley at Missouri; and Bill and Bob Jenkins at Valparaiso.
Who are next? Watch out for identical twins Lodrick and Rodrick Stewart, who wander the halls of Rainier Beach High in Seattle, daring anyone to tell them apart.
Each sophomore stands a shade under 6 feet 5, with 2 to 3 more inches projected, and each has a 40-inch vertical leap. Each is ranked among the nation's top 75 recruits. Each has a tattoo that reads "Double Trouble."
And each wants to attend North Carolina.
Two twins who didn't attend the same school were Mike Lanier, who played for UCLA, and Jim Lanier, who suited up for the University of Denver.
At 7-6, they are, according to "The Guinness Book of World Records," the world's tallest twins.
The Newton North High volleyball team that played for the Division I Massachusetts state championship this year had three sets of twins - with the twins being six of the team's top eight players.
What are the odds of that?
"It's a long shot," wrote Dan Shaughnessy of the Boston Globe. "Where are the DoubleMint gum people? This is the greatest twin story since 'The Patty Duke Show.' The Twins Souvenir Shop should make up hats and T-shirts. Two for the price of one."
In March 1993, Virginia Military Institute faced Duke in a college baseball game.
In the ninth inning, both teams had a battery consisting of identical twins: Merlin and Marlin Ikenberry for VMI and Phil and Matt Harrell for Duke.
Lefty Gomez, a star pitcher in the 1930s, rarely had success trying to retire the great Earl Averill.
"I thank the good Lord he wasn't twins," Gomez said. "One more like him would probably have kept me out of the Hall of Fame."
Basketball players who should be twins but aren't:
Ruben Boumtje-Boumtje, Duany Duany, Ajou Ajou Deng, Chy Chy Ikenokwalu and Nyambi Nyambi.
Compiled from wire reports and Web sites.