None of the four remaining teams in the NHL postseason have won the Stanley Cup many times before. Especially not recently.
Among those teams, the New York Islanders have the most Stanley Cups with four. They won those in four consecutive years from 1980-83 and reached the Stanley Cup Final in 1984, but they haven't been back since. They hadn't even been to the Eastern Conference Final since 1993.
The Tampa Bay Lightning and Dallas Stars have each won the Stanley Cup once. Dallas' came in 1999, while Tampa Bay won its Cup in 2004.