I love this discussion. Here’s how I think about it.
By fretting the string, we’re effectively shortening its length, and by bending it (in any direction), we’re increasing the tension on it, both of which increase the frequency/pitch of the note being played. Vibrato essentially increases the tension/pitch briefly, returns to the initial fretted note, and repeats however many times you bounce it. I see no way to drop below the fretted note, because you’d have to decrease the tension or lengthen the string, and the fret provides the lower limit there (as @brianlarsen has said).
As Justin says, I think the “classical” vibrato technique has some component of tightening/loosening the string tension by the friction of your finger pulling/pushing it along its length, but I also think there’s a significant component of @brianlarsen’s Hard-pressed frets at work here. It’s more difficult to keep the string pinned against the fret as you move your finger away from it, so you will naturally apply more force; that, in turn will stretch the string downward toward the fretboard. It’s the difference between these two diagrams:
and
By oscillating between these two positions, we’re combining tightening/loosening the string directly along its length and “bending” it toward the fretboard then releasing it. I strongly suspect the latter is the greater effect. We’ve all heard the change in pitch when we fret too hard or too far away from the fret.
Whatever effect we can get from “pulling” the string toward the bridge with this method is the only mechanism I can see for bring the pitch below the fretted note.