When I first got my 5DIII (brand new from B&H) I noticed the photos were not as sharp as the ones taken with my 5DII using the same lenses - a lot of back-focsing. I eventually noticed that the global setting for AFMA was set to +7. I don't recall changing that setting but may have inadvertently. I set it back to 0 and cleared everything right up (pun intended).
Recently, I used the dot-tune method to AFMA my cameras and all of my lenses individually - one of my lenses (70-200 2.
seems noticeably sharper with a +4 adjustment at both 200 and 70. The other lenses seem slightly sharper (may just be be my imagination). Regardless, it was worth an hour of my time and NO expense using that method.