Vertex Snapping doesn't "snap"?

Hello,

I've downloaded an asset from the store for caves, with modular pieces. They're not a unified size, however, so I can't use snap settings to command-click them into place. So I'd like to use Vertex Snapping.

However when I use it, I can drag the objects together (and they only move on one axis), but they do not snap when close to another. I've done this with the cave pieces and with two cubes that I just created for the test. They go through each other without a snap.

Any idea how to make this work properly? Thanks!