Voter fraud has occurred in recent presidential elections, officials in key swing states say, yet its magnitude falls short of the type of systematic cheating that Donald Trump is prophesying.

Ohio’s secretary of state discovered dozens of people who appeared to vote in two states or cast ballots as noncitizens in 2012. Pennsylvania recorded 28 fraud cases from 2000 to 2016, resulting in 17 convictions on charges such as voter impersonation and fraudulent use of absentee ballots.

In Wisconsin, officials said attempts to impersonate other voters are rare, though ex-felons sometimes manage to vote before their rights have been restored. Reid Magney, a spokesman at the Wisconsin Elections Commission, said the agency may get a couple of dozen of such cases out of 3 million votes cast.

Analysts said those kinds of irregularities, while serious, wouldn’t have swung the election in any of those states, each of which Barack Obama won by tens or hundreds of thousands of votes in both of his presidential races.

“I’m not aware of any widespread fraud in any of the key states in 2008 and 2012. Surely there was no fraud that actually made a difference to the outcome,” said Joshua A. Douglas, a law professor at the University of Kentucky who teaches and researches election law.

Questions about whether rigging or fraud could sway an election made news after Republican presidential nominee Donald Trump raised fears about it.

He has refused to agree to accept the results of the presidential election until he sees how it plays out.

Analysts debate whether his complaints are good politics. Some say it could help him drive his voters to the polls, while others say it could depress turnout all around. Those in charge of elections say Mr. Trump has overblown the dangers.

“Are there cases of voter fraud? Absolutely, there are cases of voter fraud,” Ohio Secretary of State Jon A. Husted, a Republican, told CNN. “But it’s rare, and we catch these people. Most times, we catch them before their vote is even counted and we hold them accountable, and we’re building a better system every single day.”

Mr. Husted used similar language in January 2013, when he referred 135 cases of suspected voter fraud to law enforcement after the 2012 general election.

The tally included 20 people who were registered to vote in Ohio and another state and appeared to cast ballots in both. Mr. Obama carried Ohio by more than 266,000 votes.

“This report demonstrates that voter fraud does exist, but it is not an epidemic,” Mr. Husted said at the time.

The Ohio attorney general’s office said that of the 20 people accused of voting in two states in 2012, two pleaded guilty, eight went into a criminal diversion program or other form of deferred prosecution, and six were cleared of criminal activity. Prosecutors didn’t pursue one of the cases, and three cases remained under seal.

Election integrity groups say the number of cases brought by state officials doesn’t capture the extent of fraud. Violators must be caught, for one thing, and opposition to voter ID requirements means many states don’t have checkpoints to spot bad behavior.

“States are only able to see and do what the law allows them to,” said Logan Churchwell, a spokesman for True the Vote, a group that helps people report election fraud.

New Hampshire Secretary of State Bill Gardner said at least one case of voter fraud has been prosecuted every year since 2000, though actual occurrences are hard to quantify.

“People say if someone gets picked up for DWI, they’ve driven [while intoxicated] 10 or more times,” he said.

Sometimes people are caught by chance.

Mr. Gardner noted a famous case in Manchester, where a local man spotted a voter who hadn’t lived in the area since high school.

The man, it turned out, lived in Massachusetts and traveled back to the Granite State for the 2008 and 2012 elections.

“He admitted he’d been voting up here for years,” Mr. Gardner said.

Mr. Trump also has hinted that Mr. Obama is letting illegal immigrants pour into the country to vote.

Indeed in 2012, Mr. Husted rooted out 17 noncitizens who appeared to have voted illegally in Ohio. At least four were convicted.

The 1990s-era law to push voter registration, dubbed Motor-Voter because it enlists state motor vehicle offices to dole out forms, has added to the problems.

Lloyd Leonard, a senior advocacy director for the League of Women Voters, said noncitizens who get driver’s licenses sometimes are registered to vote because of clerical errors.

“No system is perfect all the time. But it is important to know the difference between administrative errors and intentional acts,” he said. “Administrative errors do occur, but there are very, very few examples of ineligible people intentionally trying to register and vote.”

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