Last year I drew a spring bear hunt and was planning on hunting the blues (west of monticello) at around 9500 feet. I was having my doubts whether or not I would be able to hunt where I had hoped because of the heavy snow they got last year, but by the end of may I could get around. There was still some drifts on the northslope sides up high. With that being said, every year is different and from what I'm hearing up here in salt lake, that part of the state has been POUNDED so it's tough to say. Last year the snow pack was at around 120 percent for this time of year if I remember right. Go on ksl and monitor the snow pack totals (look at the la sal measurements). In about a month, it will start melting pretty good. Last year when my dad went down turkey hunting in mid april, he told me I would be very lucky to hunt my bear where I was planning because of all the snow, but by the 3rd weekend in may when we went to set my baits, we could get around (on quads) just fine. I would say don't plan on taking your 40' toy hauler up there, but cruising the roads on atv's, or trucks you'll get around. Another thing to keep in mind about that area at that time of year, it can rain 2 inches in a hour (like last year when I went back down for my hunt) and make for a "white knuckle" experience. Those roads are like snot with any significant amount of rain and you could really get yourself in a pickle. So unless you REALLY want/need to get up high, june will be your best bet. I have a good friend that is in your same dilemma because he drew the same bear tag I had last year and wants to hunt the same spot I did. Needless-to-say he is praying for a good spring heat wave.
Just be careful and use good judgement!! The high country is beautiful during the spring as it's coming out of its winter hybernation, but it can also be very brutal and un-forgiving too.