The "washington rule" that pop history gives is generally correct. there was a real strong norm against running for three terms. That being said some presidents attempted to break the mold and were defeated for a couple of different reasons. Teddy probably could have won the presidency in a two way race but Taft and others in the gop establishment were unwilling to have the sitting president bow out for a third roosevelt administration. It's a fascinating topic i can't do justice. Grant tried for a third term in 76 (more arguable but he definetly put out feelers before being rebuffed, think of mitt's noncampaign campaign this year for a weak comparison) and 80 (fractured interesting convention where grant was one of the top 3 contenders before Garfield won as a dark horse compromise candidate).

With Grant a mix of bad will towards the scandals while in office and again the very real hostility towards "caesarism" of a third term sunk his candidacy despite grant being General Grant the great war hero.

There may be more (especially since you really should consider pseudo runs as serious attempts at renomination as the "invisible primary" is a real part of the nomination process) but to take a step back some predecessors actually did run for more than two turns but they lost and a somewhat major component was the norm against running for more times reinforced in the popular press.

his successors were barred from running via constitutional amendment in 1947 (truman was exempt but both uninterested and too unpopular for a campaign against ike).

to make a brief poli sci point: it's clear that when predicting presidential elections there is at least in the modern period a "two term hangover" where you are much less likely to get a third term in a row than get a third presidential term and this plus health problems (some things i've picked up anecdotally from reading on presidents, but haven't seen anything systematic about this) does shape the general situation that makes it more likely presidents are willing to step away instead of embark on a new campaign. it's not a sexy why but the correlations are there.