Theodore Roosevelt did face some backlash when he ran again in 1912. As this cartoon https://catalog.archives.gov/id/306175 from the National Archives makes clear that although there was no Constitutional barrier to a third term there certainly was precedent. Roosevelt insisted on running in 1912 because of his disappointment with the performance of W H Taft. His complaints were numerous (see his speech given during the Republican Convention in 1912. It ends withe the memorable phrase "we stand at Armageddon...")He was defeated for the Republican nomination by Taft and quickly formed the Bull Moose Party as a vehicle to contest the election. Roosevelt's decision to fight on despite losing (or as he thought being cheated out of) the Republican nomination ripped the party in half. Woodrow Wilson won the 1912 election despite getting 42% of the popular vote. Wilson actually got less votes than some other recent Democratic Presidential candidates bit won the divided election. Interestingly Roosevelt wasn't the first President to seriously challenge the two term precedent of George Washington. Ulysses Grant had attempted to win the 1880 Republican nomination but couldn't quite capture the party after 4 years of retirement. His fight did however open the door for dark horse candidate James Garfield to win the nomination.

Are you asking in relation to the limit of 2 terms for a President?

If so, keep in mind that the 22nd Amendment which sets that limit wasn't passed until 1947, nor ratified until '51.

Taft, his opponent for the nomination, made of point of the issue saying that Roosevelt was breaking with the 2 term tradition.

See United States Congressional serial set, Issue 6177, 'The Address of the Hon. William H. Taft, section called 'The Third Term'

https://books.google.com/books?id=NN03AQAAIAAJ&pg=PP37#v=onepage&q&f=false