As noted above, if they're displayed, there's no need for a ranger even to ask. It makes her/his job easier, and (theoretically, at least), limits whatever annoyance some people might feel about being asked for paperwork in a "wilderness" settting. I don't think I've ever been checked in a National Forest, although I have been in a National Park.
In practice, the carbon copy you put in the trailhead box is likely the only thing that ever gets scrutinized.