Well, I searched a bit and have some ideas on this, but would still like to get input from those with more experience than me in this sort of thing.
Recently, I picked up a used 414ce. When I initially was playing it (along with a few others I was considering for my travel 'beater') at Empire Music, it seemed fine. As Joe and I were discussing the guitar options before me, he started noodling with the 414ce I ended up buying, and commented he thought the truss rod needed a tweak to remove a little of the relief. After he adjusted it, the action did seem better to me, the guitar still sounded great, the price was right, and the deal was done.
Knowing the 414ce was going with me to Kentucky, it sat inside its case while I played the "Wild Grain" Rosewood Custom GA I'd also purchased during my time in Pittsburgh. Then a curious thing happened last night when I pulled out the 414ce to play it: I was getting string buzz on multiple strings when fretting basic chords like C, E, and F. I think it's sometimes referred to as "fretting out." Anyway, I sighted down the neck/fretboard, found that it was still nicely straight, and the action when playing was still to my liking. But now the strings were buzzing... Puzzling...

Today, I borrowed a 1/4-inch nut driver from one of my co-workers and put some relief back into the neck (counterclockwise adjustment on the truss rod nut). But this, of course, caused the action to grow a bit higher than I'd prefer (although I can still comfortably play the guitar). And as currently adjusted, I've not totally eliminated the string buzz; but it now only occurs on the "A" string (3rd & 5th frets) when I pluck the string pretty strongly. I could add more relief to completely eliminate the buzz, no matter how hard the strings are plucked -- but then the action is getting ever higher...
Now, I've made truss rod adjustments on my other guitars -- typically in the other direction (clockwise adjustments) to remove some of the relief -- and have not encountered problems with string buzz. So now I'm left to postulate how to put the neck relief back where I'd prefer it while eliminating the string buzz...?
Being a used guitar, there's no telling what the previous owner(s) may have done; filing the string grooves in the nut, sanding down the base of the saddle...? I'm not sure if there's any way to ascertain those things by observation? Or whether the symptoms I'm describing even suggest that those things may have been done -- although I suspect that doing either/both could cause the action to be lowered and the strings to buzz when the truss rod is adjusted to factory specifications. But eliminating those possibilities for the moment, my research suggests that re-setting the neck (with different shims) may also correct the problem.
So what say you, oh forum gurus? What is the next course of action in troubleshooting this problem?