The average opening date in the last decade has been May 25th. But that's an average. It opened as late as June 4th in '05. That was another wet year. Sadly, a bulldozer operator was killed by an avalanche while clearing the road near Olmsted point on June 13, 1995. The park was cited by OSHA for safety violations. Since then YNP has adopted a much more conservative policy for opening the road.
There have also been years in which the pass was opened and then briefly closed again.While you're waiting for the snow to melt, the boulders to be blasted, and the plows to do their work, here's some
fun reading. Of course this information pertains to the entire road and is not specific to the section B/T US 395 and Tioga Pass.