The answer is that Boise State Broncos are older than the Mile High City version of the Broncos by... well... a mile.
Shortly after the start of Boise Junior College in the early 1930s, the local football team was dubbed the Broncos. And, yes, even back then the team colors were blue and orange. The Boise Junior College Broncos played their first games in 1933, when their record was 1 win, 2 losses and 1 tie. The Broncos played as a junior college until 1968, before joining the NCAA in 1969.
The Denver Broncos were founded in 1959 when Colorado was awarded an American Football League charter franchise. The team merged into the NFL, along with the rest of the AFL, in 1970. In 1960, the Denver Broncos unveiled uniforms that were white and mustard yellow. Two years later, the team colors changed to orange and blue.
Over the decades, both Boise State University and the NFL have tightened the exclusivity of their names, colors and logos, but it's a fair bet that Boise State and Denver won't be playing each other any time soon.