In 2000, a number of Americans claimed that the Supreme Court stole the election for George W. Bush. In 2008 and 2012, Obama was forced to address whether or not he was an American citizen. More recently, Trump has had to contend with claims that Russian interference got him elected.
It seems clear to me that the American people have very little confidence in the electoral process. Has this happened before in American history, where a succession of presidents have faced opposition not based on their policies but on the supposed illegitimacy of their election?
There's two 19th century elections that faced serious questions of legitimacy:
The Corrupt Bargain of 1824.
Due to the decline of the Federalist party decades earlier, the Democratic-Republican party ran 4 candidates, each nominated by different state legislatures. Although no candidate won a majority of electoral college votes, Andrew Jackson's support in the west and south got him 99 out of 261 electoral votes compared to John Q. Adams 88. Jackson also won a plurality of the popular vote.
Due to no candidate winning a majority of electoral votes, the election was decided by the House of Representatives, electing John Q. Adams, despite him losing to Jackson in both the popular and Electoral vote. The Speaker of the House, Henry Clay, hated Jackson on both a political and personal level and was accused of fixing the election. It didn't help things that Clay ended up being the Secretary of State for the new administration. Jackson and his supporters spent 4 years campaigning on accusations of Clay and Adams colluding to overturn the will of the people, leading to Jackson's defeat of Adams in 1828.
The Great Betrayal of 1877
After the Civil War, the United States army occupied several former Confederate states to help aid in "Reconstruction" and to rebuild their society without slavery. These troops and the various northern businessman, politicians, teachers and other "carpetbaggers" were unpopular with some white Southerners and many questioned the legitimacy of the new slavery-free system of laws. The instability of these new state governments led to some issues during the election of 1876.
Accusations of voter fraud, the suppression of black voters, ballot stuffing and other forms of vote tampering ran rampant throughout the south. Several states turned in multiple results, with one set of results signed by Republican (the party of Reconstruction) representatives and another signed by Democratic representatives. Some areas even managed to report more votes than there were eligible voters! South Carolina also managed to elect two governors, the incumbent Republican Chamberlain, a Union veteran and Democratic challenger Hampton, a former Confederate general. Each got about half of the recorded vote.
After a few months of arguments about which votes to count, Congress passed a law forming the Electoral Commission to settle the argument. In the informal, unofficial "Compromise of 1877" the presidency was given to Republican Rutherford B. Hayes, in exchange for various concessions to the South. Remaining federal troops were withdrawn, a new transcontinental railroad was built through Texas and newly freed blacks were no longer given federal protection.
While these might be the most notable examples, most elections have some people to question them. Coincidentally, they're usually the people who lost the election.