Severe rumbling and vibration from rear under hard acceleration
 

I have a 2004 Boxster S with 35,000 miles and recently I noticed this severe rumbling and vibration from the rear of the car when I accelerate hard from stand still. The rumbling stops when the car gets up to speed. It feels as if the rear wheels are loosing traction and are jumping up and down. The tires have 1/16" tread left. Could this be the result of shock absorbers going bad? or is this something more serious?

Well, maybe the tires are losing traction and you are experiencing something like "axle tramp." I've seen some Porsches really do quite a bit of that. Did the car break traction under hard acceleration when the tires had, say, 3/8 inch of tread left? If so, did the car vibrate like this even a bit?
If it were my car, I would replace the rear tires as a first step to resolving this prob lem. Most tires, particularly the higher performance ones, have a much softer (stickier) compound on the outer part of the tread and much harder as you get near the belts -- the compound gets much much harder (less traction) as they wear down to the final 1/4 inch or so. And face it, this really isn't costing you anything much to try --with 1/16th inch of tread you have to replace them soon regardless, 1/16th may in fact may be a legal amount of tread but I never drive a car myself with less than 1/4 inch -- too likely to hydroplane on wet roads. So, replace the tires. If that fixes it, problem gone. If not, you need new tires anyway.