Can any president win an election and then lose an election and then win an election?
Has any president ever done this in the past?
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Yes. It's even possible to be in office, lose re-election, win an election, lose re-election again and then win a second time if the first term of office results from the incumbent ascending to the presidency after the president dies or is otherwise removed from office as long as their first partial term is two years or less.
As for has it ever happened in the past Grover Cleveland lost his bid for re-election in 1888 when Benjamin Harrison won the electoral college despite losing the popular vote (this was the last time this happened until 2000 and then again in 2016). In 1892, Cleveland ran again and this time defeated Harrison in a rematch.