With rare exceptions, mid-June is the best time for riding in the Rockies. That's why one-week supported rides such as Ride The Rockies and Bicycle Tour of Colorado always pick mid-June. Your best weather in the Rockies is usually in June. Once you get into July and August, thunderstorms start posing more of a problem.
I must disagree, John. But I'll let you speak to the issue.
Here's what you said about riding Going to the Sun Road on June 24:
"For the three miles on either side of the pass, visibility dropped to near zero,
and the cycling between the 15-foot walls of snow was downright dangerous."
Ride the Rockies has considerable logistics support - plus prearranged camping and chow lines that can be moved indoors to school gyms or churches if necessary. Self-contained touring folks don't have that luxury.
In the Northern Rockies and, to a degree, the Colorado Rockies, the last big snow often occurs in early to mid June. Sure. It's gloppy and wet. And often it doesn't stick long to the roads. But for a couple of days it can be pretty nasty. (Even had a July 4th snowstorm in Jackson, WY.)
Plus, there's the issue of snow that is already on the ground and the dates when facilities open. At higher elevations, snow begins accumulating in October and piles up all winter. Real melting doesn't start until May. Campground do usually open by Memorial Day in Colorado, but in Wyoming and Montana often do not open to late June - - such as Grant Village in Yellowstone.
In 2010, I was doing census work over the summer after school got out. Here are two pictures from June 20th. The first is from town - 5000 feet. The second is where I am snowshoeing in to verify that summer houses are not permanent residences - 8000 feet. Big difference!
PS - Yes, I got paid to go snowshoeing on a glorious day.