What's the advice of great CEOs, including Vanguard Group's Jack Bogle, when it comes to admitting mistakes? Do it.

So why is that so hard?

Take, as an example, the Wells Fargo debacle, with the bank sinking deeper into a scandal over 5,300 employees who opened fee-generating accounts for customers without authorization, ultimately leading to the Senate grilling for Wells Fargo CEO John Stumpf on Tuesday. Could the Wells Fargo CEO have nipped this headline scandal in the bud if he had been more contrite?

Last week Stumpf and other senior executives first placed the blame on certain Wells Fargo employees and denied any problem with the bank's culture. "Underperformers" was the term they used to define the culprits.

By the time of his Senate testimony, Stumpf had learned it's better to hold yourself accountable. "It's against everything we stand for as a company," Stumpf said under tough questioning from senators. "That said, I accept full responsibility for all unethical sales practices in our retail banking business."

"I always thought I looked better admitting a mistake," said Bogle. It's a lesson he also learned the hard way early on in his career as a company leader.

Bogle made his biggest investment mistake when he merged the storied mutual fund company he was running, Wellington Funds, with a company in Boston that embraced portfolios heavily laden with risky equities. Wellington lost money, and the board fired Bogle.

Bogle now tells the story with a mixture of flair, humor and, still, even after all these years, a tiny bit of defensiveness. "My decisions were not quite as stupid as you might think. I was trying to save the company," Bogle said. Then he paused. "Now, that's a little bit of an excuse, and I admit that."

After that experience, he launched Vanguard and realized the only thing worse than making a mistake is refusing to admit it.

If you get it right, research suggests that the ability to admit that you are no better than the people around you, that you are humble, is the trait that sets great CEOs apart from merely good ones. Great CEOs seemed to be set on their path to greatness, according to Jim Collins, by either a near-death experience or a big public mistake.

If practiced well, admitting mistakes to yourself and to the people around you can help bring your organization together and, maybe most importantly, get you quickly to a better strategy.

Charles D. Ellis, founder of Connecticut-based consultancy Greenwich Associates and an investing industry legend himself, said, "You can say, 'I realize that a mistake has been made and it's something we've got to do something about. Or you can say, 'We've made a mistake; we're in serious difficulty. Let's gather all the smartest people around us and figure out what to do.' Either one of those can be a galvanizing experience."

So how and when do you admit a misstep?

You probably have to consult your lawyers, though you might not have to listen. Bogle said in another case, where one of his funds had a bad year, he ignored legal advice to not take responsibility in the shareholder letter.

According to business coach Keith Cupp, CEO of Portland, Oregon-based Gazelles International Coaching Association, you should also give due consideration to whom you admit the mistake: an individual, your executive team, the whole company, shareholders or another constituency.

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Last Tuesday, as public attention to the Wells Fargo scandal increased — and Stumpf's response to it proved counterproductive — he appeared on CNBC's "Mad Money" and told Jim Cramer the buck did in fact stop at his desk: "I'm the leader. I get it," he said. "When we don't meet our goals of 100 percent right, I am accountable."

Stumpf also issued the obligatory three words: "We are sorry," he said, then added, "There is nothing in our culture, nothing in our vision of values that would support that [behavior]."

During his Senate testimony — during which one senator called the bank's actions "despicable" — Stumpf added emphasis to last week's apology, saying he is "deeply sorry."

Senator Elizabeth Warren, D-Mass., went even further in her attack on Stumpf specifically, saying he provided "gutless leadership." She called on Stumpf to resign.

The Wells Fargo CEO has also seemed to be, at least in part, in job-protection mode: "The best thing I could do right now is lead this company and lead this company forward," he told CNBC's Jim Cramer.

During his senate testimony, Stumpf said, "I am fully committed to fixing this issue, strengthening our culture and taking the necessary actions to restore our company's trust."

Bogle has another guideline for separating mistakes from failed experiments, which happen often when you're building a company. "Look around and see if you've done material damage to something that already existed. "When something results in a catastrophe, it's like those people who say, 'I wouldn't have done anything differently.' ... Well, yes, if the house burned down, yes, you should have," he said.

The considerations are different, meanwhile, for women CEOs and for those working in other cultures. Women can get credit for taking responsibility, but they can also be judged more harshly for making mistakes, research from Yale University found. In the study, when a male police chief's decision to understaff an emergency resulted in 25 injured civilians, his approval dropped 10 percent. When a female police chief made the same call with the same result, her approval ratings dropped 30 percent.

Cultures can be different, too. In Western cultures, which place a high value on individual accountability, admitting a mistake can help people trust you more. "If you don't have any mistakes and you're always perfect, everybody knows you're full of B.S.," said Ellis.

The New York Times' DealBook noted last week that Berkshire Hathaway chairman and CEO Warren Buffett, which is Wells Fargo's largest shareholder, has explained the impossibility of any CEO overseeing a perfect company this way:

"Somebody is doing something today at Berkshire that you and I would be unhappy about if we knew of it. That's inevitable. We now employ more than 250,000 people, and the chances of that number getting through the day without any bad behavior occurring is nil. But we can have a huge effect in minimizing such activities by jumping on anything immediately when there is the slightest odor of impropriety."

Stumpf refused to tell Jim Cramer if he had spoken with Buffett, but that kind of rationale wasn't enough to stave off the senate grilling — Stumpf tried to employ it last week on "Mad Money."

"We have at any one time 100,000 team members in our branch and retail bank network. ... Of those 100,000, the vast majority do the right thing. ... We know running a business that not everyone is going to do everything right every minute of every day," the Wells Fargo CEO said.

In a culture that is less transparent, in which the community is more important than the individual, the idea of being set apart, not for a success but for a failure, can be unbearable, said Tim McCarthy, former chairman of Nikko Asset Management and former president of Charles Schwab. He recounted the story of an executive at Nikko, who when confronted by an error, would not admit it.

"In the U.S. it would have been a situation where I could say, 'Come on, admit it, you (screwed up),' but he couldn't get there. And eventually I had to stop, because I thought he's going to go out to the parking lot with a little white knife.'"

Why is it still so difficult for CEOs to fess up? One of the barriers that stands in the way of admitting an error is ego, said Cupp, the business coach.

"Through the media, through organizations, through careers we're trained to think the CEO is almost infallible," Cupp said. "We have spun this tale that they are infallible. It's almost a fairy tale."

Thinking over his mistake at Wellington, which led to Vanguard's founding, Bogle said he doesn't think of himself as an idiot, or a genius, either. "Know ye the truth and the truth shall make you free. I got that from the Good Book," he said. "Just think of yourself as someone who is trying to get the work done, and hope people will join you to get it done."

— By Elizabeth MacBride, special to CNBC.com