Cool article I found. They've had snow recorded in every month except maybe August?
A Memorial Day weekend storm piled three feet of snow on Whiteface in 2013, prompting some people to grab their skis.
He pointed to the writing of the explorer Samuel de Champlain, who, after exploring the lake that would eventually bear his name in July 1609, wrote that he saw mountains to the east that were covered in snow.
https://www.nytimes.com/2016/06/08/nyregion/snow-in-june-its-forecast-for-the-adirondacks.html