There's the Stanley Cup finals defeat to the Chicago Blackhawks in 2015. Then there's last May's conference finals against the Pittsburgh Penguins, in which the Lightning held a 3-2 series lead before losing two straight. And before all that, in 2014, he was the most high-profile snub of the Swedish Olympic team in Sochi, the one that lost the final to Canada.
But he's not that kind of guy.
Instead, he's trying to use the disappointments as fuel for future success.
"The first weeks after the loss were tough and I was still mad, but then, once I got back to offseason training, it was easier to start looking ahead again," he told ESPN.com recently in Swedish. "Now those losses are what's driving me, but it's a long [wait]. Also, it's a tough league, and all teams are good and getting better, so we can't get ahead of things too much, either."
If it were an episode in ESPN's 30-for-30 series, the chance for redemption would come quickly for Hedman and the Lightning. But in real life, a 10-month wait for the next opportunity is long, and if the focus is 10 months away, the chance might never come, Hedman said.
"Today, there are 30 teams in the NHL fighting for the playoff spots, and only 16 make it there. Our first goal has always been, and is now, to make the playoffs," he said. "You can't get ahead of things, but instead, our focus has to be on game No. 1, and then take it from there, and try to form good habits that carry us deep."
The Lightning still have their Stanley Cup finals core group intact, thanks to team captain Steven Stamkos' decision to re-sign with the team. Hedman sees that as an additional strength for the team.
"We have good chemistry in the group, and we have a system that we like," he said, "but I'm sure the fact that we've been through a lot together is a driving force for us all and pushes us to get better."
With the Lightning going deep in the playoffs, Hedman hasn't been able to play for Sweden at the world championships the past two years, and since then-coach Par Marts famously left him off the Olympic roster in 2014, it's been four years since Hedman has worn his country's three-crown jersey in an important tournament. That will all change when he suits up for Sweden in the World Cup of Hockey, which begins Sept. 17 in Toronto.
"I'm excited about [playing for Sweden in the World Cup] because it has been a while, even if the fact that it's been for the right reasons -- that we've gone deep in the playoffs," he said.
Sweden's head coach for the World Cup, Rikard Gronborg, spent a decade in the U.S. college system before joining the Swedish federation 10 years ago. The players, without exception, like his style, which they consider more North American than Swedish.
"He knows the American terminology and what it's like to play in the smaller rink, and he understands how big this event is in Canada," said Hedman. "He also knows a lot of [opposing] players and their styles."
Team Sweden held a preparation meeting in early August, during which the players socialized a bit and listened to their new coach's ideas and thoughts. And shared their ideas with the coach.
"The coach says how he wants us to play, but he certainly listens to our suggestions," said Hedman. "After all, he can give us templates, but we have to execute them on the ice."
With players coming from different NHL teams, and a head coach that the players call "Americanized," will there even be a Swedish style?
"You'll see," Hedman said with a grin.
"We have a lot of talent in all positions. We have a great defense, forwards that are world class, and we have one of the best goalies in the world. We'll be creative, use our defense in the offense. Then again, all teams want to play a possession game, and all teams play a similar style, with their own nuances. We just have to do it better than the other teams."
But there is one very Swedish thing the Tre Kronor players do when they get together. They do fika, which is Swedish for sitting down with a cup of coffee, talking about this and that, and just taking it easy for a while.
"We do have a lot of fika," he said. "I think I've had seven or eight cups of coffee."
That's fuel, too.