There are a number of things that can cause dead fish. Yes, you can overfeed them. If you feed them before they are hungry (about halfway down on their "Fed" meter), they will lose health each time you feed them.

Fish can also lose health and die if they are too fragile to survive in the level of Fish Environment that you currently have. Only the common fish (the ones that you can hatch from common eggs) can remain completely healthy in a Level 1 Fish Environment. The very fragile hybrids will usually not live to adulthood. Vitamins will temporarily improve their health, but only the proper Fish Environment will solve the problem.

Of course, sick fish will die if you don't cure them!

Finally, if you're playing more than one player and are changing the game speed at the same time that you're switching between players, you can have some unexpected results. Anytime you change players, you should always click play immediately. Then, if you want, you can return to the menu and change the game speed.

If none of those things explains what's happening in your tanks, maybe you could provide more details about the fish that are dying.