From the process your describing, I don't think your tank was actually cycled. For the tank to undergo the nitrogen cycle, there would have to have been a source of ammonia added. Prime itself actually binds ammonia, does not provide it. You say that first nitrate rose a bit- nitrate should actually be the last parameter to rise in the cycle. First ammonia should rise (as you add it, either in liquid form or in decaying organic matter like a shrimp), until bacteria form that eat the ammonia and turn it into nitrite. Then the nitrite should rise, until another bacteria grows that eats the nitrite, and crates nitrate. Finally the nitrate will rise and ammonia and nitrite will be at 0, indicating that the proper colonies of two types of bacteria are present in the tank to turn the ammonia (which needs to be added throughout the process) into nitrate. Simply letting the tank run for a month without constantly adding ammonia, does not cycle the tank.
So I am guessing that the tank did not have these colonies of bacteria, and through weekly water changes you kept the ammonia and nitrite low enough for the fish to last a few weeks. But eventually, the creep which happens unless you change the majority of the water in your WC, led to high enough levels to kill the fish.