In northern Wyoming the GD route goes over Union Pass and Togwotee Pass which are often snow-covered at end of May.
Snow conditions change from year to year. Some areas may have more snow than usual and others less.
In 2011 the Tour Divide Race, starting in the second week of June, had probably the worst snow conditions in its history. There were detours around the Canadian Flathead, the American Flathead and around Togwotee/Union Pass, if I remember well, and also a long snow passage in Routt NF in northern Colorado. If you take the TD-2011 route as a backup to the ACA route and decide from snowcover maps and local information which one to take, you may be able to ride the GD route from south to north without motorized transport. But be careful: it is often easier to pass over a consolidated snow cover than over freshly-cleared dirt soaked in snow melt. Sticking mud is impassable!
For the possibly blocked passes, see the chart in
http://www.bikepacking.net/forum/routes/great-divide-mtb-route-in-late-april/msg71122/#msg71122 Information about the TD-2011 detours (maps in Reply #1143) :
http://www.bikepacking.net/forum/ultra-racing/2011-tour-divide/1140/