Like Jack, I hit the JMT in June, though for me it was a little earlier, I started from Kennedy Meadows on June 7th last year, northbound. Last couple of years were relatively low-snow years, and indeed this one might be shaping up to be similar, though it's always possible more snow could come in.
I agree with Jack that it was just fine, and often fun to walk on the snow. Mostly it was hard, not something you sink into. Most thru-hikers mailed home ice axe and crampons along the way as we found we didn't need either last year. The worst part of the snow is the transition zones (snow-to-no-snow), as melting snow turned trails into streams.
Jack also said: "Of course, disclaimers apply. Don't do it if you're a newb."
Agreed. PCT thru-hikers sometimes make a bit light of what can be a lot more challenging to others, because by the time they get into the Sierras they've already come 700 miles, toughened up, gotten stronger, and figured out a lot of issues.