According to the White Mountain Research Station website
the road is in very bad shape due to runoff this year. Earlier today my wife called them (as they are having their annual open house Sunday August 7, 2011) to ask this question, since we don't have a 4WD vehicle. They said that they graded the road this year, so there aren't any big trenches any more (making 2WD possible), but that there are high rocks exposed so that a high clearance vehicle is needed. We're going to check back again before our trip up there.
We're going to drive the road as far as we can in our car (using a pick and shovel if needed), and if we have to hitch-hike the rest of the way, we will!