There are two chief considerations for a mountain trip in the west at that time of year:
1) Snowpack consolidation. Unfortunately, the snowpack isn't super consolidated yet in late may or early June. 2018 was the driest year on record in the San Juan range - got only about 50% of median snowpack. I started hiking in the San Juans on May 15, and spent a few weeks there. I was postholing by about 10am every single day, which makes for extremely slow and frustrating progress. Once you hit, say, July, all that snow will be nicely consolidated and you can walk on top of it pretty much all day. But at that time of year, you have to assume that you'll posthole most of the day. Not my vacation cup of tea, but you may be different! Slowshoes or skis would probably help, but that's all contingent on you having the training and gear of course

2) Late season snow. A few years ago, a friend and I planned a Sawtooths trip over Memorial Day. We normally wouldn't have planned something like that so early, but the Sawtooths had, again, a record-low snowpack. The week before our trip, a snowstorm moved in and put down about 2 feet of fresh up there. We had no choice but to change plans. We ended up going to Bryce and doing the Under-the-Rim trail, and had a wonderful time. Yes, it can snow in the mountains any time of year, but it's much more likely in May than it is in July, and much more dangerous in May than in July, because you can still be looking at wet-slide avalanche problems (which, by the way, formal avalanche education is a must if you're recreating in the mountains in the winter or spring).
In summary... yes, a Sawtooths trip in May is theoretically possible and, for some experienced hikers with the right training and equipment, totally achievable. But without skis/snowshoes, it's certainly going to be miserable... and you might get snowed out the week before. Just hard to know at that time of year. Agree with
@Jackson that, if you're looking to do something at that time of year, high desert is the place to be - say the upper drainages of the Escalante River, Bryce/Paunsaugunt Plateau. Even the North Rim of the Grand Canyon (course, if you go farther down, it's gonna get hot in a hurry). Or head east and do the Great Smoky Mountains!
Hoping not to dissuade you, but to put things in a realistic light. Whatever you end up doing, have fun!
