Obviously not the one solution for any time this happens, but I've come across this only once (just now), and this is what worked.
The problem was the user was copying from a Google Doc and then trying to paste into a Word doc. Nothing pasted, even though the Edit menu item flashed blue as if it was trying to paste.
I thought at first that maybe it just couldn't paste outside of Google Docs itself (has been known to happen), but the text pasted just fine into a TextEdit file.
And Word itself could copy and paste text to itself (if you typed something in the Word doc and then pasted into the Word doc, it pasted just fine).
Through just trial and error, I eventually stumbled upon the solution, which was to go (in Word) to Edit, select Paste Special, and then select Unformatted Text. The paste went through just fine.