I don't think you can count most of those National Championships. The number I can currently state that Tennessee has won without starting to giggle currently stands at 2.
The 6 seasons were 1938, 1940, 1950, 1951, 1967, and 1998.
I would not giggle so much over the 3 seasons from 1938-1940. Oddly enough, only 2 of those seasons were the Vols selected by NCAA recognized polls as being national champions. The 1939 team that did not give up a single point during the season was passed over by all polls. The Vols finished all 3 seasons with perfect 10-0 records and imho can claim championships as well as any other undefeated team. It should also be noted that bowl games were "non-games" and did not factor into National Championships until the mid-1960s. (1965 and 1968-current) The Vols final AP (pre-bowl) rank in each of those seasons was 2nd, 2nd, and 4th. However, the NCAA recognizes more than just the AP. For the 1938 season, the Vols are national champions in no less than 7 officially recognized rankings. For the 1940 season, the Vols are national champions in no less than 4 officially recognized rankings.
The 1956 team also finished the regular season with a perfect 10-0 record. They also finished 2nd in the AP (pre-bowl).
In 1967, Tennessee (9-1) finished the regular season at #2 behind Southern Cal (9-1). That was the last season when the National Championship was selected prior to the Bowl Games.
If the argument regarding bowls is made, the Vols would not have won the National Championship in 1951... BUT they WOULD have won in 1950 after defeating Texas in the Cotton Bowl. They would have also won in 1938 after defeating Oklahoma in the Orange Bowl. And lastly, you would have to go back in a time machine and tell the coaches and players that the game actually mattered and was not just an excuse to take a trip.
However, to sum up, according to the website, College Football Data Warehouse, the most acceptable selectors throughout history are the National Championship Foundation, Helms Athletic Foundation, and the College Football Researchers Association. The Vols are noted for 4 National Championships amongst these few rankings. They were for the following seasons: 1938, 1950, 1951, and 1998.