(CNN) - Barbara Bush made national headlines last year when she said on NBC that her son Jeb shouldn't run for president in 2016 because "there are a lot of great families. It's not just four families, or whatever. There are other people out there that are very qualified. We've had enough Bushes."
And in an interview earlier this year on C-SPAN, the former first lady added that "I think this is a great American country, and if we can't find more than two or three families to run for high office, that's silly."
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A new national poll indicates that most Americans agree with Bush, the wife of former President George H.W. Bush.
Sixty-nine percent of people questioned in a NBC News/Wall Street Journal poll released Wednesday said they agree, with a quarter saying they don't agree with her statements.
With both former Florida Gov. Jeb Bush and former Secretary of State Hillary Clinton seriously considering bids for the White House in 2016, the discussion of political dynasties is alive and well. If Bush ran and won in 2016, he would be the third Bush in the White House over the past three decades, following his father (1989-1993) and his brother George W. Bush (2001-2009).
If Clinton ran and won the next election, she would be the second President Clinton in the White House in the past two decades, following her husband Bill Clinton (1993-2001).
While Barbara Bush doesn't want Jeb to follow in her husband and son George's footsteps, it's not because she doesn't think he's up to the task.
"He's by far the best qualified man, but no, I really don't" want to see him run, she has said.
Jeb acknowledges dynasty issue
While Jeb Bush has joked about his mother's comments, he's also discussed how his family name may hurt him as much as help him if he decides to run for the White House in 2016.
Speaking in February at the Long Island Association's biannual luncheon, a popular stop for former presidents and White House hopefuls, Bush was asked about his mother's remarks.
"It's an issue for sure," Bush admitted.
Bush told the story of sitting next to a man on a plane who talked about having a Bush and a Clinton and then a Bush in the White House, with the prospect of another Clinton and Bush to come.
Bush said "I get the point. It's something that, if I run, I would have to overcome that. And so will Hillary, by the way. Let's keep the same standards for everybody."
George doesn't see eye to eye with Barbara
Jeb Bush's father may not agree with his wife.
In an exclusive interview with CNN Chief Political Analyst Gloria Borger, Jeb's brother, Neil, made it clear that his dad - George H.W. Bush - wants his son to run, no matter what his wife says.
"If you asked Dad the same question 'Should Jeb run?' he'd say yes." Asked again whether Bush 41 would specifically be for it, Neil Bush replied, "Yeah, he would say yeah."
Has Neil asked his father himself? No. But he allowed, "I have heard him answer that question."
Hillary's numbers higher than Jeb's
The NBC News/Wall Street Journal poll, like previous surveys, finds Clinton more popular with the public than Bush. Clinton has a 48%-32% favorable/unfavorable rating in the new poll, compared to 21%-32% for Bush.
Previous polling from other organizations also indicates Clinton leading Bush, as well as other potential Republican White House hopefuls, in hypothetical 2016 general election showdowns.
CNN special coverage of the 2014 midterm elections
The NBC News/Wall Street Journal poll was conducted April 23-37, with 1,000 adults nationwide questioned by telephone. The survey's overall sampling error is plus or minus 3.1 percentage points.