I finally got time to try the attacks again, and they worked without problems. I rebooted both the "attack" and the "victim" and tried again, and again it worked.
I honestly don't have a clue why it didn't work several times that I tried it before, but I guess that comes with the territory. Nothing is guaranteed to work.
The only notable difference between the failed runs and the successful runs was that on the successful run, I forgot to enter the WEP key into Ettercap.
At any rate, it works now