After overtaking Hillary Clinton in some Iowa and New Hampshire polls, Sen. Bernard Sanders is showing he has a real chance of winning South Carolina and crashing through the Clinton campaign’s supposed Southern firewall, analysts and Democratic Party leaders say.

Should Mr. Sanders seize momentum once voting starts in the Democratic presidential nominating contests, there’s a growing belief that Mrs. Clinton’s 36-point lead in South Carolina could quickly evaporate, just as it did in 2008 when then-Sen. Barack Obama swamped Mrs. Clinton en route to the White House.

While there are many key differences between 2008 and this presidential cycle — not the least of which is Mr. Obama himself, who eight years ago was able to galvanize African-Americans, young voters and other key groups in the party like few before him — Mrs. Clinton’s campaign seems to once again be on the ropes, and a loss in South Carolina could spell doom for the former secretary of state.

“I think there could be a meltdown,” said David Woodard, a political science professor at Clemson University. “I think South Carolina could be very critical. If Bernie wins the first two, he might win here I wouldn’t be surprised if Bernie Sanders was able to knock her off here.”

Democratic leaders in the state believe the momentum is flowing toward Mr. Sanders. Wins in the Iowa caucuses and New Hampshire primary, they say, could dramatically change the South Carolina race overnight, and caution against assuming that Mrs. Clinton’s massive lead will hold once voters begin heading to the polls.

“I absolutely believe that can happen, and the reason I believe it can happen is because I witnessed it here in 2008,” said Allen Bailey, chairman of the Sumter County, South Carolina, Democratic Party and a Sanders backer.

He added that many supporters of Mrs. Clinton, at least in his county, seem ripe to be pulled to the Sanders side provided the senator spends more time in the state.

“I think Bernie’s biggest challenge is himself. He’s just not as well known,” Mr. Bailey said. “He’s had to introduce himself down here. All the people out front for Hillary they’re not necessarily avid Hillary supporters. They’re settling for Hillary.”

Mrs. Clinton’s 2008 South Carolina loss was the combination of momentum growing behind Mr. Obama — who won the Iowa caucuses several weeks earlier, a feat once considered virtually impossible — and self-inflicted wounds, the largest of which were gaffes uttered by her husband, former President Bill Clinton.

Mr. Clinton said, among other things, that Mr. Obama’s candidacy was little more than a “fairy tale.”

So far the Clinton campaign has avoided those kinds of glaring missteps and continues to enjoy a huge lead in South Carolina and a substantial lead nationally.

A RealClearPolitics average of all South Carolina polls conducted between Dec. 5 and Dec. 17 show the former secretary of state with a whopping 40 percent advantage over Mr. Sanders.

The RealClearPolitics average of all national Democratic primary polls gives Mrs. Clinton an 8-point advantage.

But the Clinton campaign also seems keenly aware that the race’s momentum may be shifting in Mr. Sanders’ favor, evidenced by polls showing the senator ahead in both Iowa and New Hampshire.

Clinton campaign leaders over the past several days have launched a concerted effort to undermine Mr. Sanders in the minds of progressive voters by targeting his stances on gun control and the details of his health care plan.

“One can only conclude the Sanders campaign does not want to outline what is going to amount to a massive, across-the-board tax hike for working families. They would prefer to avoid that,” Clinton campaign adviser Jake Sullivan told reporters on a conference call Wednesday afternoon to discuss Mr. Sanders’ health care platform. “They want to essentially create a circumstance in which they try to lead voters to believe they can implement single-payer health care at no burden to anyone, and everyone will be better off.”

Mr. Sanders told CNN Tuesday night he will release the details of his plan, and how it’s paid for, before the Iowa caucuses on Feb. 1. His campaign also called the Clinton campaign’s attacks “false and misleading.”

The Sanders campaign did not respond to questions about its plans in South Carolina and whether it will beef up staffing in the hopes of cutting into Mrs. Clinton’s lead there.

Meanwhile, the Clinton campaign insisted this week that it’s not attacking Mr. Sanders solely because of the changing polls. Campaign officials stressed that they’re still confident about Mrs. Clinton’s prospects in Iowa and New Hampshire.

“We’re confident in the ground game we have built in Iowa and New Hampshire,” Clinton spokesman Brian Fallon told reporters this week.

In South Carolina, however, party officials say polls showing Mrs. Clinton with a seemingly insurmountable lead are, to some extent, worthless because many voters haven’t truly made up their minds.

“A fair amount of them are waiting to see how things are going to shake out,” said Harold Crawford, chair of the Aiken County Democratic Party.

On campus at Clemson University, Mr. Woodard says he sees Sanders supporters handing out stickers and other campaign paraphernalia on a daily basis. He says he’s seen virtually none of that from Clinton backers.

“She just doesn’t turn you on,” he said of Mrs. Clinton and why Democratic excitement is once again behind another candidate.

But some South Carolina Democratic leaders say excitement and momentum won’t be enough for Mr. Sanders, even after wins in Iowa and New Hampshire. The key to a victory in South Carolina, they say, will depend on whether his grass-roots foot soldiers will work as hard as Mr. Obama’s backers did in 2008.

“If Bernie were to pull something like that off, that’s how it would be done. It wouldn’t be an invisible thing called momentum,” said Amy Hayes, chair of the York County, South Carolina, Democratic Party who worked for Mr. Obama’s campaign in 2008 and has remained neutral so far in this cycle. “It would be his supporters — are they motivated enough to get out and knock on doors and talk to people? That’s what we did for Obama.”

Other Democratic leaders across the state argue Mrs. Clinton’s support is stronger than it was in 2008, and voters that may have left her side eight years ago won’t do so again.

“I think it’s solid,” Newberry County, South Carolina, Democratic Party chairman and Clinton supporter Dave Waldrop said. “I’m sure human nature would kick in, and some people would say she can’t win. But I don’t believe it would be a domino effect. But I could be wrong; anything could change.”