Most paved passes will be open by early June. Might be cold at the top, but the highways will usually be plowed by then. That said, if a front goes through you could get snowed on. Maybe even snow packed roads on the high passes for a day. But it won't last long. Just expect it could happen and delay your travels for from a few hours to a day or so. The dirt passes on the other hand probably won't be open. Even passes like Trail Ridge Rd are usually open for Memorial Day. Often it closes again for a few hours or even a day. Beartooth may be a bit iffy in early June, but Chief Joseph should be open to Cook City etc. The Yellowstone Plateau is higher than most people think. And you get more moisture on that side, so weather is more severe.