Cannot Update iMac, OS X, From 10.11.2 to 10.11.6
I'm not sure what's going on with the App Store update, but I have a guess about the downloaded update: did you get the "OS X El Capitan 10.11.6 Update" (759.1 MB), or the OS X El Capitan 10.11.6 Combo Update (1.5GB)? If you downloaded the first one, it won't work because it's only capable of updating from 10.11.5 to 10.11.6; since you have 10.11.2, you need the "combo" version, which rolls in all of the earlier 10.11.x updates, and is therefore capable of updating from any earlier version of 10.11 to 10.11.6.

(BTW, this requirement is listed on the download pages for the two updates, but you have to click "System Requirements" to actually see it, and even then it's not very explicit. The first says it requires "OS X El Capitan 10.11.5", but the second just says "OS X El Capitan 10.11".)