It's happened several times. Most recently in 1980 when Ted Kennedy challenged the incumbent President Jimmy Carter. Kennedy fought all the way to the convention.
The last sitting Republican President to be seriously challenged was Gerald Ford in 1976 by . . . future President Ronald Reagan. Reagan also fought all the way to the convention.
Notably, in both of these cases, the challenged President managed to earn his Party's nomination but was unsuccessful in the general election.
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That does happen, and it's not as rare as I thought. Those who do challenge an incumbent President for the party nomination are usually not a serious threat, but there have been a few challenges in the past half century worth mentioning. For example, in 1976, Ronald Reagan competed against incumbent President Gerald Ford for the Republican Party Nomination. That race was too close to call right until the Republican National Convention, where Ford narrowly won the nomination. He lost the election, however. Also, in 1992 Pat Buchanan ran against incumbent President George H. W. Bush for the Republican Nomination. 73% of Republicans voted for Bush in the primaries. In 1980 Ted Kennedy (the U.S. Senator from Mass. who died in 2009 and brother of the former President) competed for the Democratic Nomination against incumbent Jimmy Carter. Although Carter had 24 Primary wins to Kennedy's 10, Kennedy refused to concede until he lost the nomination in a 2129 to 1146 vote at the Convention. Many were surprised when Eugene McCarthy ran against Lyndon Johnson for the 1968 Democratic nomination. Four years earlier Johnson had the highest percentage of popular votes of any U.S. presidential candidate since George Washington. After Johnson received only 49% of the vote at the New Hampshire primaries to McCarthy's 42%, Robert Kennedy also entered the race against Johnson. It became obvious to Johnson that the Democratic nomination was something that he was going to have to work for, but all his time was consumed by the war in Vietnam as well as the urban racial unrest domestically, so he withdrew from the election at the end of March 1968. Pete McCloskey and John Ashbrook challenged Richard Nixon for the 1972 Republican nomination. Out of 1324 delegates to the Republican Convention, Nixon won 1323 and McCloskey won 1.
Yes, they can be challenged, but only for the nomination. A political party nominates only one candidate for president and all party members do their best to support the candidate of their party. ( A person can desert his party and run against his old party's candidate, as did Theodore Roosevelt in 1912. )
A Democrat could run for the Democratic nomination against an incumbent Democratic President. This happened in 1980 , for example, when Ted Kennedy ran against President Carter. A democrat would not run on the same party lines as another Democrat since the Democratic party can only nominate one candidate themselves. So a third party could nominate a Democrat to run for president if that Democrat lost the Democratic nomination to someone else.
The 22nd Amendment to the US Constitution limits a US President to two elected terms in office. However, he may also serve up to 2 years of another President's term. However, if he has already served more than 2 years, he is limited to one elected term.
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