In her 27 years running elections in New Bedford, Maria Tomasia says, she can think of one voter who tried to game the system. A man posed as his father and cast a ballot because the polling location was closer to his work.

But as Republican presidential candidate Donald J. Trump ratchets up unsubstantiated claims that next month’s vote will be rigged, Tomasia and other state election officials are defending the integrity of the system that will pick the next president.

“It’s very disheartening,’’ Tomasia, chairwoman of New Bedford’s Election Commission, said Tuesday. “These issues always come up just before the election and there really is no evidence that this ever happens.’’

In recent days, Trump has repeatedly warned the vote will be fixed, telling a rally in Wisconsin Monday that ballots are being cast in the names of deceased voters and illegal immigrants. On Twitter, he accused Republican leaders of ignoring “large scale voter fraud.’’

His comments were echoed by a key supporter, Republican Maine Governor Paul LePage, who said Tuesday that ‘‘people from the cemetery’’ and some who aren’t citizens will cast ballots because identification isn’t required to vote in his state.

“We need to have a photo ID when we go to vote,’’ he said. Until that happens, ‘‘I don’t think the elections in the United States or the state of Maine are legitimate.’’

Secretary of State Matthew Dunlap, Maine’s top election official, disputed LePage’s contention. ‘‘The accountability procedures we have in place now make any attempt at wide-scale manipulation of an election mathematically impossible,’’ he said.

Studies examining voter fraud have found a few minor cases, none of them serious enough to affect the outcome of an election.

One researcher, Loyola Law School professor Justin Levitt, found only 31 instances of alleged voter fraud between 2000 and 2014, when more than 1 billion ballots were cast.

Secretary of State William F. Galvin, the top election official in Massachusetts, said this week the system is layered with protections.

Votes are cast on paper ballots, which are saved after the polls close, and candidates can request a recount by submitting signatures from voters, he said. Following the election, 3 percent of precincts will be audited. Wednesday was the deadline in Massachusetts to register to vote in the Nov. 8 election.

“There are many safeguards to make sure the votes actually cast by voters are counted and tabulated correctly,’’ Galvin said.

When officials suspect fraud, they address it quickly, he said. Galvin cited the case of Enrico J. Villamaino III, a former selectman in East Longmeadow, who was convicted in an election fraud scheme.

Villamaino and his girlfriend at the time, who was a volunteer at the East Longmeadow town clerk’s office, changed the registration of more than 280 voters from Democrat to unenrolled to make them eligible to vote in the GOP primary for state representative in 2012, then forged the unsuspecting voters’ names on absentee ballot requests, prosecutors said. The ballots were never mailed.

“We caught it,’’ Galvin said. “Once we figured out what he was doing, we referred it to the [district attorney].’’

In Boston, Election Commissioner Dion Irish said his staff spends weeks testing optical scanner voting machines and backup equipment. Boston police officers are also assigned to polling locations, where they help verify the identity of voters and maintain order, Irish said.

The officers are also responsible for delivering ballots to polling locations on Election Day and, once polls close, returning them to City Hall, where they are stored in a locked vault, Irish said.

“We put a lot of effort into training. We’re always updating our voter list,’’ he said. “We have a very solid operation in place.’’

Mary Audley, city clerk in Lynn, said she was not surprised by Trump’s comments and has heard similar claims from unsuccessful candidates.

“But I have never had one come up with any evidence of fraud,’’ Audley wrote in an e-mail. “We take a person’s right to vote very seriously. We also want to make sure that their vote counts and that everyone who is eligible to vote does, and those that aren’t don’t.’’

Andrew T. Dowd, president of the Massachusetts Town Clerks Association, declined to comment on Trump’s remarks but said clerks are committed to elections that are “conducted fairly and in compliance with all laws and that every ballot cast is counted and recorded correctly.’’

In North Andover, where voters backed former Republican Governor Mitt Romney for president in 2012, Town Clerk Joyce Bradshaw said the system is sound.

Voter rolls are checked against government databases to look for duplicate registrations, and death records are reviewed to remove deceased voters from lists, she said.

“When you sign that paper certifying the vote from . . . your community, you’ve put your personal integrity and that of your community on the line,’’ Bradshaw said. “That’s the most important thing we do.’’

Across state lines in Salem, N.H., Town Moderator Christopher Goodnow said it was “ludicrous’’ to suggest that election results in his community could be vulnerable to tampering.

Trump won the Republican primary there earlier this year.

“I’m not concerned about them being rigged at all,’’ Goodnow said.

Trump was also victorious during the Massachusetts Republican primary in which a record 637,703 voters cast ballots for GOP candidates, Galvin’s office said.

“He won the Massachusetts primary pretty overwhelmingly,’’ Galvin said. “He didn’t seem to have any problem with our process.’’

Material from the Associated Press was used in this report. Travis Andersen of the Globe staff contributed. Laura Crimaldi can be reached at laura.crimaldi@globe.com.