Here is the example that got me thinking:
How could they inset those pieces? I examined one of these in real life and it was very convincing. I couldn't find seams or joints or anything.
This is what's called a bandsaw box, and this particular one was probably created using roughly the steps below, which are slightly different from those for a more typical bandsaw box.
It appears there's also a secret compartment in the bottom of the large 3-compartment drawer. That secret compartment was probably cut out after step 2, then the secret drawer itself was created using a similar process:
Note that in step 1, you typically don't cut off the front of the log or blank if you are going to have one or more drawers that are visible from the front; instead you cut off the front after you cut out the drawer(s).
You may have noticed from the sequences of steps above that there are basically 4 cuts that you can mix and match in different orders to come up with pretty much any bandsaw box design: cut off the back, cut off the front, cut out the drawer(s), cut out the compartment(s) in each drawer. Once you know this, you can also look at any bandsaw box and work backwards to figure out how it was made.
I couldn't find seams or joints or anything.
Because the bandsaw blade's kerf is so small, it's possible to close up any entry cuts or other seams so that they are practically invisible. You can also strategically make cuts with the grain to make them even less visible.