There usually is very little snow there by the end of June. I would say the trails are usually snow free by about the end of May. But we had a crazy winter here in So NV in 2005 with record rains in Vegas and huge snows in the mountains.
After hiking Mt. Charleston countless times in warm weather, I did my first winter climb up through the Big Falls area last April. I'm hooked and can't wait for the snow to come back so I can do it again!
"So I was sitting in my cubicle today, and I realized, ever since I started working, every single day of my life has been worse than the day before it. So that means that every single day that you see me, that's on the worst day of my life."
--Peter Gibbons (Office Space)