"Our country is now looked upon as the foremost warlike nation on Earth, and there is almost a complete dearth now of commitment of America to negotiate differences with others," Carter said.
And throughout much of Clinton's presidency, Carter was a Democratic agitator. He embarked on his own foreign policy projects, at times at direct odds with the White House, and after Clinton's affair with Monica Lewinsky became public, Carter questioned Clinton's morals.
Even after Clinton left office, Carter continued to criticize.
In February 2001, a month after the Clinton presidency had ended, Carter said Clinton's last-minute pardon of Marc Rich was a "most serious mistake" and even alleged that the reason Rich was pardoned was donations that Rich had made.
"I don't think there is any doubt that some of the factors in his pardon were attributable to his large gifts," Carter said. "In my opinion, that was disgraceful."
But Carter has been the exception, not the rule. While some presidents have stepped out and criticized during the heat of a political campaign, it is a rare occurrence.
"Carter puts issues ahead of fraternity and will often speak out about things," Gergen said. "I think it is part of his nature. Maybe being a one-term president changes your view."
'Little Boy Blue' and the Bay of Pigs
Dwight Eisenhower is by most accounts the gold standard for how to deal with other presidents.
Eisenhower chewed out his successor, John F. Kennedy, in private for the botched Bay of Pigs invasion, but publicly offered him a level of support.
Kennedy had been in office for just three months when he called for a meeting with Eisenhower in April 1961 after the attempt to topple the young Castro regime in Cuba was a massive failure. The operation had originated before Eisenhower left office and Kennedy's administration had carried it out.
The Kennedy-Eisenhower relationship had been frosty. Eisenhower had long viewed Kennedy as naive and young -- he referred to Kennedy as "Little Boy Blue" in private -- and Kennedy had disparaged the Eisenhower administration in his 1960 presidential campaign.
The two put their chilly relationship aside, however, and Kennedy invited the former general to Camp David to review the mistakes in Cuba.
According to Eisenhower's notes from the meeting and a number of media reports, the conversation between the new and former president previewed the pitfalls of foreign policy campaign promises:
Kennedy: "No one knows how tough this job is until he's been in it a few months."
Eisenhower: "Mr. President. If you will forgive me, I think I mentioned that to you three months ago."
After the meeting, Eisenhower stood in front of television cameras and said, "I am all in favor of the United States supporting the man who has to carry the responsibility for our foreign affairs."
In "The Presidents Club," Gibbs and Duffy write that despite the fact that all of these men have risen to the highest level of their profession, they not only work better together, but they need one another.
"There is no fraternity like it anywhere, and not just because of the barriers to entry of the privilege of membership," they write. "For all of the club's self-serving habits and instincts, when it is functioning at its best, it can serve the president, help solve his problems and the nation's, and even save lives."