Instead of modifying and save as copy, you might get better results if you go to Windows Explorer, copy and paste your part, re-name the new one, and then modify it as needed.
On the other hand, in the situation you describe I believe I would use configurations instead of seperate part files.
I agree with Glenn. I would use the configuration route. This way you can insert the second one and if you RC on it and select properties, that brings up a dialog window that allows you to select the configuration you want.
I would also in your case use a design table to adjust all of the different sizes.
In the part window you make a config that is the whole part at the original size, than a config for each size that you need.
The trick is to (create an assembly from part) for each config state and clearly label the created assembly. You can then create a new assembly and drop in all different sizes of bearings you want, all referencing a single part in the part window.
This allows for the base part to be changed and to have the change propagate through all the models or apply just to the config you have open. When changing dimensions you have option for this config or all configs.