Overfishing
Overfishing can exhaust the fish population of a pond or river, both temporarily and permanently. If a dwarf can not find some fish in a specific area, you will get the message "there is nothing to catch in the [directions] swamps" and the area will no longer be used for fishing. When the season changes, all areas will become available for fishing once again, though eventually they will remain permanently empty and simply give the above message on the first fishing attempt of every season.
http://dwarffortresswiki.net/index.php/FishingSome fish are vermin, others are catchable, apparently.
If you set a fishing zone, and set orders to only allow in-zone fishing, you're safe from overfishing.
If you don't set zone-only fishing, and have several fisherdwarves, their skill will increase, making them fish faster... if you ignore them running all over the map to grab the last turtle, they will completely deplete aquatic wildlife. No more fishies.
... I'd suggest switching to farming.
To get shell for moods, just import turtles and lobsters, and forbid turtle and lobster from being cooked.