I guess mid June is close to that huh?
Weekends in June in the cascades often suck ass. I'd say you are lucky to get one decent weekend in June, especially on Rainier. Last year, I had two weekends set aside for Fuhrer Finger and both fell through due to weather. But, there was perfect weather up there one weekend at the end of May (memorial day weekend, I think).
If you can do midweek though, and run out with little notice, well your odds are better.
The weather does seem to stabilize the weekend *after* the 4th of July.