I'll change what I said earlier to 1824. Jackson won the popular and electoral vote yet didn't win presidency. Bloody ridiculous (not that I want him as president as he was an HP) but he clearly won that round and Quincy got elected because of congress.
Jackson did not win the electoral vote. If he had, he would have been President.
Calhoun did win the electoral vote, and he was elected Vice President.
The 1824 election was a bizarre intra-party election. The electors from Connecticut (who were popularly chosen) voted for Adams for President and Jackson for Vice President.