Does increasing concentration of copper sulfate decrease or increase its conductivity?
I'm thinking maybe at the higher concentrations your tap water is making a film of $\ce{Cu(OH)2}$ on the surface of your electrode, and that is what's shutting down your electroplating. Your tap water might be mildly alkaline (tap water pH's can range from about $6.5$ to as high as $9.5$).

Try adding a little sulfuric acid to your solutions. If alkaline films are the problem, that will fix it. 

It will also increase the conductivity of your solutions tremendously, because hydrogen ions are extremely mobile. 

The solubility of copper sulfate decreases with increasing sulfuric acid concentration, so don't add too much acid.