Catcher. Gary Carter was an 11-time All Star with the Montreal Expos and New York Mets for most of the 1970s and 1980s. He led the NL in RBIs in 1984, and piled up a 70.1 career WAR across 19 seasons. For his accomplishments Carter was elected to the Hall of Fame in 2003.
Yet he never did manage to claim an MVP. Carter came close twice, finishing second behind Mike Schmidt in 1980 and third in the Mets’ 1986 championship season behind Schmidt and Houston first baseman Glenn Davis.
First base. Over a 23-season career mostly as a force in the Cincinnati lineup, Tony Perez created a Hall of Fame resume. He hit .280 with 379 home runs and 1,652 RBI. Perez hit 40 home runs with 129 RBI in 1970, his third of seven plus-100 RBI seasons. With a career WAR of 54.0, he won election to the Hall in 2000.
But Perez never had any luck with MVP voters. Somewhat amazingly, he only received MVP votes in seven of his 23 seasons, coming no closer than third (behind teammate Johnny Bench and Billy Williams) in that brilliant 1970 season.
Second base. Craig Biggio spent his full 20-season career with the Houston Astros, earning seven All-Star selections. He piled up 65.4 career WAR, four times topping .300 in batting average and .400 in on base average.
Yet somehow, Biggio only really gained MVP support twice. In 1997, he finished a distant fourth behind Larry Walker, Mike Piazza and teammate Jeff Bagwell, and, one year later, he was fifth behind Sammy Sosa, Mark McGwire, Moises Alou and Greg Vaughn.