"The Rose Bowl was the only major college bowl game in 1930. By 1940, there were five major college bowl games: the Rose Bowl, the Sugar Bowl (1935), the Cotton Bowl Classic (1937), the Orange Bowl (1935), and the Sun Bowl (1935). By 1950, the number had increased to eight games. In 1960 there were still eight major college bowl games, but by 1970 the number had increased again, to 11 games. The number continued to increase, to 15 games in 1980, to 19 games in 1990, 25 games in the year 2000 and as of 2010[update], 35 games in total. Up until around the 1950s, games were played solely on New Years Day, with few exceptions. In the late 1950s, more bowl games began playing their games earlier in December. Also bowl games began to be set in cities which were not thought of as winter vacation destinations due to their colder climates."
BYU was pretty fortunate that they became a good program around 1980 when the amount of bowl games had increased almost four fold from 1940. During Utah's hey day from 1920 to 1960 it was almost impossible to be invited to one of existing bowl games. Most of them were more exclusionary than the current BCS, which only allowed conference champs from a conference that Utah was not a part of.
To gauge Utah's success prior to 1970 based off of the number of bowl games they played in wouldn't be very prudent.