The answer to this question is inevitably the Leafs, who managed to lose 5 Cups in 6 years in 34-35, 35-36, 37-38, 38-39, and 39-40.
Bob Davidson, Red Horner, Pep Kelly and Nick Metz were on all 5 teams.
Buzz Boll, Busher Jackson and Bill Thoms were on 4 of the 5, but only lost 2 in a row, though neither Boll nor Thoms played in the 38-39 playoffs.
Reg Hamilton was also on 4 of the 5, but lost 3 in a row.
Syl Apps, Murph Chamberlain, Gordie Drillon, Bingo Kampman, and Turk Broda were on all 3 consecutive loss teams.
Charlie Conacher was on 3 teams but only lost 2 in a row.
Pep Kelly won 0 cups, Red Horner 1, Bob Davidson 2, Nick Metz 4.
Buzz Boll and Bill Thoms won 0 cups, Busher Jackson 1.
Reg Hamilton won 1 cup.
Bingo Kampman and Gordie Drillon won 1 cup, Murph Chamberlain 2, Syl Apps 3, Turk Broda 5.
Charlie Conacher won 1 cup.
Charlie Conacher, Red Horner, Busher Jackson and Bill Thoms were also on the 32-33 Leafs team that also lost in the final, to go along with Andy Blair, King Clancy, Hap Day, and Joe Primeau who added a 3rd finals loss to their resume.
Andy Blair, King Clancy, Charlie Conacher, Hap Day, Red Horner, Busher Jackson, Joe Primeau were all on the 31-32 Leafs team that actually managed to win a final.