It's not so cut and dried. There is room for mischief. From Wikipedia:
Under the Twenty-second Amendment, the President of the United States may not be elected to more than two terms. However, there is no similar such limitation as to how many times one can be elected Vice President. Scholars disagree whether a former President barred from election to the Presidency is also ineligible to be elected Vice President, as suggested by the Twelfth Amendment. The issue has never been tested in practice.
To be clear, here’s the last sentence of the 12th Amendment:
“But no person constitutionally ineligible to the office of President shall be eligible to that of Vice-President of the United States.”
That seems more clear than “suggested”.