A more pertinent example might be the 1948 election when Southern Democrats abandoned the Democratic Party because Civil Rights was part of the platform. They formed the States Rights Party, also known as the Dixiecrats. The nominee was Strom Thurmond, who carried five southern states. The end result was that Truman carried enough electoral votes to deprive Thomas Dewey of the election and deliver re-election to Harry Truman. Truman had been predicted to lose; in fact newspapers had already predicted his loss. Had it not been for the Dixiecrats, Thomas Dewey would have been President of the United States.
The most famous instance of this in any recent election came in the 2000 presidential election. The third party candidate did not get a meaningful number of votes in the country as a whole, but he arguably decided the election even so. This third party candidate was Ralph Nader, running as a member of the Green Party.
This was the election that came down to the contested recount in Florida. George W. Bush was named winner by a margin of only roughly 450 votes. At the same time, Nader won more than 97,000 votes in the state. It is reasonable to assume that Al Gore, the Democratic candidate, would have won the vast majority of those votes because Greens are more likely to vote Democrat than Republican if they have to.
In this election, then, the presence of the Green candidate likely tipped the election in Florida and, thereby, in the whole country.