loose back wheels

While I was changing the back brake pads on my '98 Blazer, I noticed that I could grab the wheel and move it in and out at least 1/16th of an inch.  This seems like a lot of play in the axle.  Am I right, or is this normal?  Also, I've had my wheels balanced twice and I still get a bad shimy over 60 mph.  I thought this might be a symptom of the problem.  Any and all help would be appreciated!

My Bravada had this same play in it, but seemed like more than a 1/16 of an inch.  I didn't have any problems though. 

Have you had your u-joints replaced recently?  Your vibration could be caused by bad u-joints or having the driveshaft out of phase.  If you have recently had the u-joints replaced, try unbolting the driveshaft at the rear axle and rotating it 180*.

The play in the axle is normal. That is the "c" clip in the axle groove and the diff side bearing clearances combined.
 
Did you have all 4 wheels balanced?
Where do you feel the shimmy?
 
If you feel it in the steering wheel, it's coming from the front. If you feel it in the seat or floor, it's from the rear.

*Moving to Steering, Suspension, & Drivetrain*

I'm moving this to the Steering, Suspension, & Drivetrain section where others can find it should anyone be asking the same question.