I still don't really know what the light grey boxes around the blobs are supposed to represent. The manual says "These invariably lie directly on the semitone and coincide exactly with a gridline." But they quite clearly do not coincide "exactly with a gridline" in many cases. Not even in the image the used to illustrate them with in the manual. In many cases they're just way off. For example in one of my own projects:
 

 
And it doesn't matter what I have the grid set to: all the way up to 32nd notes, those grey boxes are not coinciding with the gridlines. The manual also says  they "are also the positions they will snap to if you double-click on them with the timing or pitch tools," but this clearly isn't the case either in the case of the timing tool. So based on what the manual is telling me, they don't work as described and I really have no idea what the point of them is. Any clues? 

I'm probably not following exactly what you're asking but I believe it's relative timing. If you move the blobs they'll snap by not to. You have to break the link between them if you want to align them on the grid. There's a Celemony video called vocal timing that may discuss what you're asking. http://www.celemony.com/en/melodyne/real-use

I get the snap by thing, but I have a Melodyne clip in front of me now in which I've gotten the blobs snapped exactly to the grid, yet the "show intended notes" boxes still extend a little way past each gridline. I guess when I read things like "these...coincide exactly with a gridline" in a manual then I'm apt to take that literally and wonder why it isn't the case in reality. 

You're thinking of the wrong grid line. They lie on the semitones - i.e. A, A#, B, C etc.

This is the row grid, if you like, not the column one.

They show the intended pitch of the note, not its position in time.

HTH.

Free tutorial on time handle tool...
http://www.groove3.com/str/player.php?id=3640&productid=16289