In this case the reason why clip doesn't work is invalid geometry. As I mentioned in the comment before:
The clip should definitely give you clipped features from layer A (smaller polygons - DouglasCountyNE2) within layer B (1 big polygon - sitearea2).
There are two reasons I can think of why you get empty result. First is that the layers are not in the same projection. You can Save as... both layers into the same projection just to make really sure (In this particular case the layers are i the same CRS so problem is elsewhere). The second is invalid geometry in one or both of the files. And that is the case here - layer B (1 big polygon - sitearea2) has 2 geometry errors.
In QGIS you can find out through Vector / Geometry tools / Check geometry validity... and then try to manually replace the errors or use some of the tools intended for removing these kind of errors.
Generally on of the simplest ways is to run from Advanced interface of Processing /Toolbox tool called GRASS - Vector - v.clean. You can leave default cleaning tool, GRASS should take care of all basic geometry errors anyway.
In the new QGIS 2.12 there is also new internal plugin called Check Geometries which after turning it on in Plugins / Manage and Install Plugins... you can use from Vector / Geometry tools / Check Geometries (please be aware at this moment after activating the plugin there are two items with the same name Geometry tools in the menu - seems like small QGIS bug).