In testing my lenses on my new Nikon D7000 (serial 3021xxx), I've discovered a weird failure.
In live view mode, auto focus on the 80-400mm 4.5-5.6 VR Nikkor (VR-off, locked down on Gitzo 3541XLS) lens fails to focus in the range of 200+-400mm. Oddly, the auto focus rectangle is green indicating it has locked focus, but the image is obviously very blurry, even when pointed at a LensAlign target. The range from 80 to 200 mm focuses just fine in live view. The 80-400mm auto focuses fine over the whole range of 80 to 400mm through the viewfinder so the lens is working correctly. The live view focus problem also happens outside where there is plenty of light, so this is not an inside lighting problem.
My 80-200mm f2.8 Nikkor auto focuses in live view just fine over its entire range. In a spot test at the zoom range limits only, the following Nikkors also work: the 18-200mm AFS VRII G DX, the 28-200mm f3.5-5.6G, the AFS 24-85mm f3.5-4.5G, and the 50mm f1.8D.
A problem similar to the 80-400mm reported above is observed with the Sigma 10-20mm does not focus with live view over any of its range. Unlike the 80-400mm problem which amazingly shows a green rectangle and a blurry image, live view displays a flashing red rectangle when the Sigma 10-20mm is focused. The Sigma 10-20mm focuses sharply when focused through the viewfinder so there is no problem with the Sigma lens.