We were up in Big Bear last weekend. A pretty cool XC ride went on with some stream crossing, snow patches here and there plus some soft dirt in areas.
The ski resort area still has snow on the forest service land - it's in the trees and you can see it from the main drag.
So i'd say almost any time for XC non lift riding.
I did run into Shannon at the hardware store (the guy in the ticket sales booth) he said he's catching up on *spring cleaning and such* before he goes back to the shack in a few weeks. Not sure if bikes will be allowed up this early in the season though.
It is still pretty cool up there though. Water for the dogs froze outside the cabin Sat night.
July - Sept is ideal for the camping and biking