Twenty-second Amendment to the United States Constitution
Some argue that the 22nd Amendment and 12th Amendment bar any two-term president from later serving as vice president as well as from succeeding to the presidency from any point in the presidential line of succession. Others contend that the original intent of the 12th Amendment concerns qualification for "service" (age, residence, and citizenship), while the 22nd Amendment, concerns qualifications for "election", and thus (strictly applying the text) a former two-term president is still eligible to "serve" as vice president (neither amendment restricts the number of times an individual can be elected to the vice presidency), and then succeed to the presidency to serve out the balance of the term (though prohibited from running for election to an additional term).
I do not choose to run
Some feel that Coolidge was honoring the two-term tradition set by President Washington. While Coolidge held the opinion that a vice president who succeeded to part of another president's term would not violate the rule by then running twice for election in his own right, this tradition was continued through him and in 1952 and 1968, by Presidents Harry S. Truman and Lyndon B. Johnson when the latter chose not to seek another term after disappointing showings in the New Hampshire primary. Both had succeeded to the presidency after the death of their predecessors and had been elected only once. Theodore Roosevelt also did not run in 1908 after his succession to the presidency in 1901 and his election in 1904. However, he ran unsuccessfully in 1912, against his successor, William Howard Taft.