Yes, but water has surface tension. Haven't you ever read about that before? Think of it like the difference between sticking your finger in the water and slapping the flat of your hand on the surface of it. The latter causes a lot more resistance, doesn't it? Surface tension will tear the paper faster than just getting it wet will. And the rim of the net is a lot sturdier than the paper.

[There does seem to be something in what Claude's saying, at least; this new approach does seem to have gotten Dimitri's net into the water without instantly tearing.]