We were also there the first week of June this year. There was a grad night. Many (most?) of the major entertainment offerings weren't running.
That's the bad.
The good: small crowd levels! Last trip we went a week earlier(which was Memorial Day weekend that year)...and the crowds were twice the size! Other than a large tour group from Israel that seemed intent on beating everyone in the HM line black and blue with their VERY large and very pokey backpacks, everyone was friendly and quite polite.
We didn't run into any wild children, rude adults, or fighting families. We spent 5 days in the parks and never once ran into a school group. There was a HUGE family...all wearing lime green...but they traveled in smaller groups. It was a never a problem. Maybe it was the combination of the good weather and the short lines!
Would I go again that week? HM. I'm not sure we're going to go again at all...mainly because Disney makes it impossible to plan ahead for a trip to DLR. I MUCH prefer WDW where I can make my plans and count on them to have their entertainment up and running.
My advice would be: as long as you'll be there for one full weekend, do it. The 4th - 11th would probably be a great time. The crowds got heavier on Saturday...heavy enough that if I were going an extended time and could plan an "off" day, I'd plan it for Saturday- or maybe just go to the park that night for shows! But, compared to Memorial Day weekend, the first week of June was a dream!