It's widely reported that the 1960 election was stolen by Democrats in favor of JFK. I don't know that it's widely disputed much anymore.
Many people believed that Kennedy benefited from vote fraud, especially in Texas, where his running mate Lyndon B. Johnson was Senator, and Illinois, home of Mayor Richard Daley's powerful Chicago political machine. These two states were important because if Nixon had carried both, he would have earned 270 electoral votes, one more than the 269 needed to win the majority in the Electoral College and the presidency. Republican Senators such as Everett Dirksen and Barry Goldwater also thought that vote fraud "played a role in the election", and that Nixon actually won the national popular vote. Republicans tried and failed to overturn the results in both Illinois and Texas at the time—as well as in nine other states. Some journalists also later claimed that mobster Sam Giancana and his Chicago crime syndicate "played a role" in Kennedy's victory in Illinois.
Nixon's campaign staff urged him to pursue recounts and challenge the validity of Kennedy's victory in several states, especially in Illinois, Missouri, and New Jersey, where large majorities in Catholic precincts handed Kennedy the election. However, Nixon gave a speech three days after the election stating that he would not contest the election. The Republican National Chairman, Senator Thruston Ballard Morton of Kentucky, visited Key Biscayne, Florida, where Nixon had taken his family for a vacation, and pushed for a recount. Morton did challenge the results in 11 states, keeping challenges in the courts into the summer of 1961; however, the only result of these challenges was the loss of Hawaii to Kennedy on a recount.
Kennedy won Illinois by less than 9,000 votes out of 4.75 million cast, or a margin of 0.2%. However, Nixon carried 92 of the state's 101 counties, and Kennedy's victory in Illinois came from the city of Chicago, where Mayor Richard J. Daley held back much of Chicago's vote until the late morning hours of November 9. The efforts of Daley and the powerful Chicago Democratic organization gave Kennedy an extraordinary Cook County victory margin of 450,000 votes—more than 10% of Chicago's 1960 population of 3.55 million, although Cook County also includes many suburbs outside of Chicago's borders—thus barely overcoming the heavy Republican vote in the rest of Illinois. Earl Mazo, a reporter for the pro-Nixon New York Herald Tribune, investigated the voting in Chicago and "claimed to have discovered sufficient evidence of vote fraud to prove that the state was stolen for Kennedy."
In Texas, Kennedy defeated Nixon by a narrow 51 to 49% margin, or 46,000 votes. Some Republicans argued that Johnson's formidable political machine had stolen enough votes in counties along the Mexican border to give Kennedy the victory. Kennedy's defenders, such as his speechwriter and special assistant Arthur M. Schlesinger, Jr., have argued that Kennedy's margin in Texas (46,000 votes) was simply too large for vote fraud to have been a decisive factor. Russell D. Renka, a former political science professor at Southeastern Missouri State University, acknowledged that it was more than likely that Johnson's political machine in the state's lower Rio Grande Valley counties, including the notorious Duval County, could have "managed to produce a significant number of forged votes" for Kennedy. However, Renka also acknowledged that Kennedy's margin in the state's initial tally "made it far too difficult to prove that voter fraud had determined who won Texas" and that "any recount would also have been hard to conduct."
Cases of voter fraud were discovered in Texas. For example, Fannin County had only 4,895 registered voters, yet 6,138 votes were cast in that county, three-quarters for Kennedy. In an Angelina County precinct, Kennedy received 187 votes to Nixon's 24, though there were only a total of 86 registered voters in the precinct. When Republicans demanded a statewide recount, they learned that the state Board of Elections, whose members were all Democrats, had already "certified" Kennedy as the official winner in Texas.
In Illinois, Schlesinger and others have pointed out that, even if Nixon had carried Illinois, the state alone would not have given him the victory, as Kennedy would still have won 276 electoral votes to Nixon's 246 (with 269 needed to win). More to the point, Illinois was the site of the most extensive challenge process, which fell short despite repeated efforts spearheaded by Cook County state's attorney, Benjamin Adamowski, a Republican, who also lost his re-election bid. Despite demonstrating net errors favoring both Nixon and Adamowski (some precincts—40% in Nixon's case—showed errors favoring them, a factor suggesting error, rather than fraud), the totals found fell short of reversing the results for either candidate. While a Daley-connected circuit judge, Thomas Kluczynski (who would later be appointed a federal judge by Kennedy, at Daley's recommendation), threw out a federal lawsuit "filed to contend" the voting totals, the Republican-dominated State Board of Elections unanimously rejected the challenge to the results. Furthermore, there were signs of possible irregularities in downstate areas controlled by Republicans, which Democrats never seriously pressed, since the Republican challenges went nowhere. More than a month after the election, the Republican National Committee abandoned its Illinois voter fraud claims.
However, a special prosecutor assigned to the case brought charges against 650 people, which did not result in convictions. Three Chicago election workers were convicted of voter fraud in 1962 and served short terms in jail. Mazo, the Herald-Tribune reporter, later said that he "found names of the dead who had voted in Chicago, along with 56 people from one house." He found cases of Republican voter fraud in southern Illinois, but said that the totals "did not match the Chicago fraud he found." After Mazo had published four parts of an intended 12-part voter fraud series documenting his findings which was re-published nationally, he said, "Nixon requested his publisher stop the rest of the series so as to prevent a constitutional crisis." Nevertheless, the Chicago Tribune (which routinely endorsed GOP presidential candidates, including Nixon in 1960, 1968 and 1972) wrote that "the election of November 8 was characterized by such gross and palpable fraud as to justify the conclusion that [Nixon] was deprived of victory." Had Nixon won both states, he would have ended up with exactly 270 electoral votes and the presidency, with or without a victory in the popular vote.
Anecdotally, campaign pollsters I've talked to on both sides of the aisle say that any GOP lead needs to be above the margin for error (about 3-5%) for them to be confident of a win, as that is about the amount of fraud committed in any given election.