Third week of June should normally be okay for the Half Dome cables to be up.
This past year they went up during Memorial Day weekend. They were down before I started my hike to Merced Lake and were up by the time I circled around and climbed Clouds Rest and Half Dome.
Two years ago they were up in early May due to the very low snowfall that winter during the drought. They won't be that early this year... 100% guaranteed.

Here's a table of the opening dates for the last 7 years:
Opening Dates (look under Trails section)So in 2011 and 2010 the cables were up by late June. I think the next winter was the first signs of the drought, so with the large snowfall we've had already, late June is likely this year.
But you may have to cross your fingers and hope for the best.
I have a trip planned for the 4th of July weekend that includes climbing Half Dome at the end, so I'll be hoping right along with you.
