How many shims are too many shims

It will still be fine being a 2 by 6. It will be in solid contact with the floor everywhere so a 2 by 6 would be plenty. It would even be ok being a 2 by 4. The top frame is more important because it isn’t in solid contact all the way around. It has to be able to support the wieght inbetween the vertical uprights.
 
The bottom frame is really just there so you have something to attach the vertical posts to. It's (usually) not load bearing and dimensions can be chosen out of convenience, within reason.
 
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