While there are some autofocus changes between the D3/D3s and the D4 you can pretty much use the same settings as you did with your D3/D3s and achieve good results on the D4. Since you focus using the AF-On button (like I do) I'm assuming you set Custom Setting a4 to AF On Only. I achieve sharp focus with all my lenses using the D4 and a couple of primes that required AF Fine Tune on my D3 do not require any adjustment on my D4. My percentage of in-focus shots with my D4 is close to 100% and far higher than with any other body I've shot with.
So what you are experiencing does not sound normal, nor does it sound like user error. I think your D4 needs a trip back to the mother ship.