And some women are asking: Would a male candidate be asked to defend his likability? What does likability signify in a candidate, anyway? And can a female candidate even afford to be likable, without compromising the need to appear tough and competent?
As for Clinton herself, she tried to blunt the question with a little rueful humor before launching into her own defense. "Well, that hurts my feelings," she said. "But I'll try to go on."
It was an approach that impressed Daniela Ioan, a mother of two, who watched the debate on replay the next night at her home in Hamden, Conn. In fact it was the debate, during which she felt Barack Obama and John Edwards were teaming up harshly on Clinton, that made Ioan decide to support her.
"She's not measured by the same yardstick," says Ioan. "Even if you don't like her, she shouldn't be scrutinized the way she is. I felt like I had to do something to help her."
The likability question also rankles Kate White, editor in chief of Cosmopolitan magazine. "I think (it's) totally unfair," White says. "She's brilliant, hard-working, energetic, clearly a great mother because she raised a dazzling daughter, and by the accounts of many people who know her, funny and warm. I don't see what's not to like."
And feminist icon Gloria Steinem feels the issue smacks of "a double, maybe triple standard."
"There is still no 'right' way to be a powerful woman - and almost no wrong way to be a powerful man - so the standard for likability in powerful men is incredibly low, and the one for women is incredibly high," Steinem wrote in an e-mail to the AP. "It's impossible unless you happen to be powerful in a 'feminine' role that no man could occupy - for example, Oprah."
Of course, male candidates are expected to be likable, too. Bill Clinton was masterful at it, and President Bush - as Sen. Clinton has herself pointed out - was known as the candidate people wanted to have a beer with. (Al Gore, on the other hand, was seen as stiff and not likable enough.)
But the difference is that these men could appear likable with no threat to their image of toughness and ability to lead. In other words, they could laugh, be empathetic, hug, kiss, and yes, even cry, without seeming less capable of having their finger on the nuclear button.
By contrast, when Clinton's eyes moistened in her famous "moment" with a New Hampshire voter - a moment many feel helped lead to her surprise victory there - Edwards had this response: "I think what we need in a commander in chief is strength and resolve."
It's all emblematic of a longstanding double bind for female candidates, says one specialist in political communication.
"It's gender bias, plain and simple," says Kathleen Hall Jamieson, director of the Annenberg Public Policy Center at the University of Pennsylvania. "They're supposed to be warm and accessible, because that's what's perceived to be gender-appropriate. But they also need to be tough and competent. The minute they appear that way, their warmth and accessibility are called into question."
She notes that Geraldine Ferraro, the 1984 Democratic vice presidential nominee, was asked if she was tough enough to press the nuclear button. That was almost a quarter-century ago. "It should be easier by now for a women to certify their credentials as being tough and competent but also warm and caring," Jamieson says.
There are, of course, parallels in the corporate world. It's no accident, says Mary Trigg of the Institute of Women's Leadership at Rutgers University, that only 2 percent of CEOs at Fortune 500 companies are women.
"Research demonstrates that there's a very narrow bandwidth of acceptable behavior for women in positions of power, because they have to be feminine, but also exhibit the kind of attributes we look for in a leader," says Trigg, director of leadership programs at the institute.
"A female leader can't be too assertive and strident, because you get into the 'B' word," says Trigg (a word that a supporter of Republican Sen. John McCain famously used not long ago about Clinton). "It's a hard and lonely trail to walk."
At least one expert doesn't see Clinton's "likability problem" as a gender issue.
Rather, says Steven Cohen, professor of public administration at Columbia University, it's that Clinton's straightforward, no-nonsense style pales in comparison to the often inspirational rhetoric coming from the charismatic Obama. "It's not her, it's him," Cohen says. "She's run into a political tsunami."
But, says Cohen, "Hillary and Bill are right in saying the media is holding them to an unfair test. Who else is being asked the likability question? Is Rudy Giuliani being asked that question?" (The former New York mayor, he notes, is a subject of a book, "Nasty Man," by fellow former mayor Ed Koch.)
Some people, asked about the likability question, wonder aloud how it ever became a factor in how we choose a president. Cohen says the phenomenon dates back at least to Franklin D. Roosevelt. "He had a wonderfully reassuring radio voice," Cohen says.
White, the Cosmopolitan editor, attributes it to the proximity we now have through the media to politicians, and everyone else in public life.
"So we have to come insist on mediagenic qualities from not only politicians but also from actors, TV hosts, even book authors," White says.
She thinks it's indeed possible for powerful women to be both tough and likable. But politics, she says, is "so much trickier. There's no blueprint for (Clinton) to follow, and it has to have been so tough to know how to play it in various situations - like when you are standing on a stage with nine guys in power suits who are all hoping to see you fail."
