A third of a millimeter is pretty thin though I just looked at a piece I am working on that is a tenth of a millimeter. A good laser cut alone is about a quarter millimeter but then what I am working on has a lot of work ahead. Another already done has a lot of places ~1/100th of an inch or a quarter millimeter because I made it 8" square instead of 4".
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The really shocking thing is that 1/72" piece comes out in one piece, if I give the hardwood time to cool. If the cuts happen right after another, it has more char and is more fragile.
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The tiger is 
medium walnut so that is very resistant, If I made the tiger in zebrawood it would be eaten by the laser.
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Still tricker is this tiger in light cherry and only 4.5 inches high and wide it is too delicate and was all I could do to weed it on one side to be able to apply the paper at which time it was a hair easier to weed. In many cases an entire section is held in place by a series of 0.01" links and no secondary so just waving in the wind. I tested it in cherry so I will do what it takes to finish it but I went in and beefed up several places and made so anything complicated had at least two connections.
A note on weeding:
The key fact is that the masking sticky side will stick to itself far stronger that it will stick to anything else, even if the connection is less than a square millimeter.
This also includes the wood material. A secondary fact is that a very tiny pull will still pull the masking from the wood however slowly. Therefore, even if the masking is a quarter millimeter wide and will break if pulled too hard, it can be pulled below its breaking point and eventually pull cleanly away.
This means that even if hard to get started, if the tiniest bit can be pulled free and facing up, another piece of masking sticky side down will not grab anything hard except that tiny bit of flipped masking and it will grab that very well. If the piece is engraved with lots of tiny islands the sticky side down masking will pull the edge of that island just enough to grab the whole piece and remove it. If the piece is pierced as with the tiger the entire masking could be pulled as one piece if one is careful and patient enough. I was not patient enough for this piece but do that with slightly less challenging pieces all the time by just keeping up a very slight and steady pull.
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