Technically, Pitt has never won one of the crystal balls footballs it displays in the montage; that trophy didn't exist until 1986, a decade after Pitt's last national championship claim of any kind, in 1976. As a matter of fact, you'll only find three, at most four, national championship trophies in the Panthers' trophy case, a MacArthur Tophy, a Grantland Rice Award and an AP National Championship Trophy from 1976, and maybe an AP trophy from 1937, the second year of the poll's existence (it's not clear whether there was a trophy in those days, what with the poll's infancy and the Depression and all). Those are the two years Pitt has actually been voted and recognized as the national champion by contemporaries.
As for the others, not only is there no trophy, but there was no vote, nor, you know, any concept of a "national champion" by the teams themselves. According to the NCAA's count, the Panthers wereretroactively awarded the 1910, 1915, 1916, 1918, 1929, 1931 and 1936 titles by at least one of several backward-looking agencies and computer systems like Richard Billinglsey, the National Championship Foundation and Dunkel Index that filled the historical void in lieu of polls. Pitt didn't lose a game for the entire span of World War I -- even by Europezn terms -- but there was no one around to recognize it until years later.
I'm not sure how many of these extremely mythical championships the school officially claims, but from the looks of it, they've updated every single one -- mythical, shared, backward-looking and otherwise -- into modern Waterford crystal form for the benefit of viewers during extra points and field goals. Because 100-year-old, retroactive tradition is still tradition.