Nemo: "I can do this, Dad!" Can he?
In "Finding Nemo," a huge school of large fish are caught in a net. They escape when Nemo instructs them to all swim downwards, overloading and breaking the fisherman's crane, which projects out horizontally over the water. Assuming fish would do this, would it work? Give the fish every benefit of the doubt. This is, after all, a kids' cartoon.
As you know, fish have a neutral density. They weigh almost exactly the same as the water they displace. When you pick up an object under water, it seems light because the water tries to float it, and nothing lighter than water has any apparent weight at all. When the fish push up against the water to flee, they are lifting the water and reducing its weight, so all the apparent weight added by the fish swimming down is counteracted by the reduced weight of the water they are pushing upward.