I would say even HOF QBs are part of the problem when the team is not good enough. And defining a HOF QB without considering the team around him is impossible.
Look at John Elway. Elway went 14 years not being good enough to get his team over the hump. Critics could point to him having 21 TD passes to 18 interceptions in his first eight playoff seasons as part of the problem. Then, suddenly Denver won back-to-back Super Bowls and Elway is a surefire Hall of Famer. (OK, he would have made it anyway but had no Super Bowl titles to hang his hat on.) Elway didn't suddenly become a different QB — he threw 6 TDs to 3 ints. in the playoffs during those Super Bowl runs. Instead, the team around him got better, particularly giving him a load-sharing running game that was one of the best in the league.