Bode Miller, Meghan Duggan, Julia Mancuso

Although skiers Bode Miller, left, and Julia Mancuso, right, appreciated the fact that they received bronze medals for their efforts at the Sochi Olympics, Meghan Duggan and her women's hockey teammates felt disappointment in taking the silver. (Getty Images / February 22, 2014)

SOCHI, Russia — Twelve hours wasn't nearly long enough to ease their pain.

The morning after losing in overtime to rival Canada, the U.S. women's hockey players talked about the anguish of walking away from the Sochi Olympics with silver medals around their necks.

"We didn't train as hard as we did for second place," forward Meghan Duggan said. "It's one of those things we have to swallow."

Contrast her words with the sigh of relief from Bode Miller a few days earlier. The ski racer acknowledged making a big mistake on the super-G course, saying he felt "really lucky" to hang on for bronze.

FRAMEWORK: Best images from Sochi

With one day remaining in these Winter Games, the Americans are second in the total medal standings at 27, but gold accounts for only a third of that total.

The rest of the medalists have landed a step or two lower on the podium, 11 of them on the bronze level. And, for the most part, their experience in Sochi has raised a curious possibility.

Finishing third might feel better than finishing second.

"With silver, you're just one person away," said skier Julia Mancuso, who has medals of every color from the last three Olympics. "But with bronze, it's like, I could have been totally off the podium."

More than a century ago, in "The Principles of Psychology," William James wrote about the paradox of "a man shamed to death because he is only the second pugilist or the second oarsman in the world. That he is able to beat the whole population of the globe minus one is nothing; he has 'pitted' himself to beat that one; and as long as he doesn't do that nothing else counts."

Researchers from Cornell and Toledo examined this notion after the 1992 Barcelona Games, studying the emotional reactions of silver and bronze medalists on the podium and in post-competition interviews.

They were testing the concept of "counterfactual thinking" which — in the Olympics — can work two ways.

"Silver medalists may torment themselves with counterfactual thoughts of, 'If only…' or 'Why didn't I just,'" the study concluded. "Bronze medalists, in contrast, may be soothed by the thought that, 'At least I won a medal.'"

One of the researchers, Scott F. Madey, now a psychology professor at Shippensburg University in Pennsylvania, said: "It runs counterintuitive to the idea of medal standings, one, two, three."

Gymnast McKayla Maroney might be the poster girl for this theory. At the 2012 London Olympics, she fell on the vault and settled for silver, her smirk prompting a breakout series of Internet memes.

At these Games, many of the top American athletes have fallen short in other ways.

Lindsey Vonn did not even make it to Sochi, unable to recover from a knee injury, and Shaun White failed to defend his snowboard halfpipe title. Four-time medalist Shani Davis came up empty in long-track speedskating and Kikkan Randall, considered a top contender in women's cross-country, lost in the quarterfinals of the sprint event.

Given those disappointments, not all of the silver medalists were unhappy.

"It's the best feeling in the entire world," slopestyle skier Gus Kenworthy said after sharing the podium with two other Americans. "I don't even have words to say."

And not all bronze medalists were thrilled with their result. Hannah Kearney, the defending champion in moguls, spoke of being "downgraded to third place."