Offhand, I can think of a few cases where we were close to having a President and an Anti-President;
1) 1800: Originally, electors voted for two candidates each, and the overall winner became President and the runner-up became Vice-President. However, political parties and tickets quickly moved into play, resulting in parties putting up two candidates, although it was known which was the "real" Presidential candidate and which was the Vice-. However, the electors voted for both candidates of a party, resulting in a tie between Jefferson and Burr. While it did take 36 ballots in the House to elect Jefferson, Burr is reported to have not tried to win the Presidency.
2) 1876: While Tilden won the popular vote, he was one electoral vote shy, with 20 votes in four states being disputed. Basically, a deal was cut that made Hayes the President, but his administration stayed under a cloud because of it.
3) 1960: There's reasonable evidence that Nixon was robbed due to vote fraud led by LBJ in Texas and Mayor Daley in Illinois/Chicago. Supposedly Nixon declined to strongly challenge this since it'd damage the office of the Presidency for whoever won.
4) 2000: As Al Gore puts it "I used to think that you either won or lost an election, but it turns out there's this little known third possible result". This one was so close that I personally consider it a statistical tie, but again Gore did not go rogue when his opponent was declared the winner. I sometimes wonder what might have happened if this had occurred in 2004 instead, when it'd become clearer just how awful a President Bush is.
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1) 1800: Originally, electors voted for two candidates each, and the overall winner became President and the runner-up became Vice-President. However, political parties and tickets quickly moved into play, resulting in parties putting up two candidates, although it was known which was the "real" Presidential candidate and which was the Vice-. However, the electors voted for both candidates of a party, resulting in a tie between Jefferson and Burr. While it did take 36 ballots in the House to elect Jefferson, Burr is reported to have not tried to win the Presidency.
2) 1876: While Tilden won the popular vote, he was one electoral vote shy, with 20 votes in four states being disputed. Basically, a deal was cut that made Hayes the President, but his administration stayed under a cloud because of it.
3) 1960: There's reasonable evidence that Nixon was robbed due to vote fraud led by LBJ in Texas and Mayor Daley in Illinois/Chicago. Supposedly Nixon declined to strongly challenge this since it'd damage the office of the Presidency for whoever won.
4) 2000: As Al Gore puts it "I used to think that you either won or lost an election, but it turns out there's this little known third possible result". This one was so close that I personally consider it a statistical tie, but again Gore did not go rogue when his opponent was declared the winner. I sometimes wonder what might have happened if this had occurred in 2004 instead, when it'd become clearer just how awful a President Bush is.