In modern history, only one President to date (Bush 41) has run for reelection but was defeated. His fellow Texan, fellow Navy veteran, fellow Republican, and fellow wealthy conservative Ross Perot had expressed, inter alia, great disappointment with President Bush regarding the terms of the Bush negotiated and implemented North American Free Trade Agreement (NAFTA).
Perot predicted that the great swishing sound you would be hearing would be as a result of NAFTA jobs (particularly Texas jobs) flowing into Mexico. To show his disgust, Perot withdrew from the Republican Party, established a new (third) party, and announced he would run for President against President Bush as the standard bearer of the new party.
In the 1992 presidential election, Perot drew a lot of conservative votes from President Bush, and even some from Democratic candidate Bill Clinton: 18.97% of the votes cast. But Clinton won more votes than President Bush, thanks to Perot, and won the presidency with a plurality (less than half), not a majority, of the votes.
Separate but related, in the 2000 presidential election, Bush’s son won the election and became President George Bush (Bush 43). This was the third time in USA history that two Presidents have come from the same family line: the others were Adams and Roosevelt.