The 61 focus points on the 5DIII are a mixed bag of different double cross, single cross, and non-cross types. This is an important distinction. A focus point is actually a small line, or group of lines. And best focus is achieved when whatever you are focusing on crosses one of those lines at or near a right angle. So if a non-cross-type focus point you are using is oriented horizontally and you are trying to focus on something that has primarily a horizontal orientation in the frame, there is very little intersecting cross section for the AF point to latch on to.
Like wise if the AF point is vertical and you are trying to focus on something that is standing straight up and down.
Again, best results are achieved when the AF point orientation is at or near 90-degrees to whatever you are focusing on.
A cross-type AF point combines AF lines in two directions perpendicular to each other - usually horizontal and vertical. In the 5DIII, there are also dual-cross points that use diagonal crossing lines.
Of the 61 AF points in the 5DIII (and 1DX) system, only 41 are cross-type sensors. The remaining 20 are single-line AF points.
I have mine configured to disable the non-cross-type AF points. That's one of the settings in the AF configuration system. Trying turning off the non-cross-type points and see what you get when you know you are always using a cross-type AF point regardless of camera orientation.
There's a fabulous guide to the 1DX AF system and the only difference between it and the 5DIII system is the AF-point linked metering on the 1DX. Otherwise they are identical so this guide is equally valuable and pertinent to 5DIII users.http://www.learn.usa.canon.com/resources/articles/2012/1dx_guidebook.shtml