I recently repacked the bearings on my open car hauler. After approximately 350 miles the wheels have some play/end shake. Just enough to feel, maybe a 0.010 inch or so. Is this good or should there be no perceivable play?

 

There will be a little play when you shake the wheel. If you tighten them to the point of zero movement the bearings will burn out from overheating. Sounds like they are O.K. to me. Check them in a few more hundred miles and re-tighten them if they get looser.
Joe

 

That's correct, they should have a little play.

Dorf

 

What I do is put them on where the wheel spins just barely, back them off a 1/2 a turn or so, now it has "some" resistance left. Put everything back together and take a short ride. Then jack the trailer up, if the wheel does not spin freely and make many revolutions, then the bearing is too tight.

During my first haul, I pull over and feel each hub/wheel after 10 miles or so to make sure it is not hot. Then I check them whenever I stop.

To add to what everyone else has said, the other thing about overtightening is depending on design, even if you back off, you can destroy the grease seal too