Yes; it was part of a general project aimed at preventing popular movements from any sort of lasting power in American government. The idea that FDR was breaking any sort of tradition by running for a third term is entirely false and easy to debunk as well. The 22nd Amendment was an entirely novel idea.
No idea how much of this is true, but a point in its favor worth mentioning is that America, beyond the first thirty years or so, never really had regular two-term presidents until after the 22nd Amendment was passed. This isn't cause and effect--party rotation became more regularized, and leaders started living longer. The only three I can think of that served a full eight years and nothing but are Jackson, Grant, and Wilson, and at least one of these had third term aspirations. America was a nation of one-term, assassinated, or accidental presidents, combined with either quick party rotation in office or long periods of one-party presidential rule.
Madison could've easily run again in 1816, if you want to go far back, and given the mortal weakness of the Federalist Party, he would've won easily if he did. Madison lived a GOOD long time in his post-presidency and was in much better health than the other ex-Presidents ended up being in.
Of course, Madison was hardly possessor of a...flawless...Presidency, but then again, who can judge what we would 've done against the might of the British Empire in the 1810s?
Anyway, I think out of the early Presidents, he's far and away the most likely to try. Washington and Monroe were in poor health on leaving their second terms and Jefferson...very clearly was OK with moving on. He just wouldn't have tried.
