A lot of people start the TransAm in VA in early to mid-May to beat the heat/humidity in the Midwest and reach the western mountains when the weather is likely to be much better (i.e., low chance of pass-closing snow and cold temps). With that said, anything can happen during those transitional months. A few days before I headed out on the Trans Am from Missoula in mid-June of 2014 there was 5" of wet snow on Lost Trail Pass. It was gone by the time I got there two days later. As for the Northern Tier, Logan Pass in Glacier National Park is not likely to open until at least mid-June, if not a week or so later, and you wouldn't want to miss that ride. It all depends on the snow levels this winter. Yellowstone can also be quite cold at night in June due to the altitude. (Old Faithful is around 6,000' above sea level.) One night there was frost on the ground near the end of June. The nice thing is that if it's sunny, things warm up pretty quickly.