I went to Montana State University. It is a good college, and I would recommend it to anyone. However,that is the only reason to live in Bozeman. If you come to college, you would be wise to live and work on campus the entire time. Here is why:
Bozeman has an insanely high cost of living. Keep in mind that most people are working very low-paying jobs and working 2 or 3 of them. If you are a college student, you will have a hard time finding a job off campus, most people will not hire students. Job competition is fierce because many students hang around town years after they graduate.
Unless you are making a minimum of $100,000 a year, you will probably never afford a home in Bozeman, maybe in Belgrade, probably in Manhattan or Three Forks. Renting is also very expensive. Most rentals cost as much as a house payment in most cities. Many of the landlords in town are horrible slumlords. Most will not rent to college students and many of those that do are some of the worst slumlords. I knew people who had property that ended up being condemned.
In general, the people in Bozeman are very unfriendly. Many of them have a better-than-thou attitude about themselves and have no basic manners. As mentioned above, the town has a very anti-student attitude.
Traffic is also obscene for a town this size. There was absolutely no planning and the growth is concentrated on one street that in a matter of 10 years has had almost every big box store imaginable built on it. People usually drive well under the speed limit and there is traffic every hour of the day as many people living here are trust fund babies who don't work.
Bozeman is a beautiful town, it is nice to visit, but I would recommend only doing that. As a single person it was bad enough, I cannot imagine what it would be like to raise a family there.