Thank you EllenD for sharing your experience and insight with this. Wow, what a scene to walk into. I will remember that.
This is called "Community Breeding" and it's highly advised against, and I can speak from experience that you never, ever, ever want more than one breeding pair of any species of bird in one cage with one nest box. Not ever...
I used to breed English and American Budgies, along with cockatiels and several conure species...well due to a space constraint, years ago I built a huge indoor aviary in my basement, larger than any flight cage you could possibly have, it took up half a large room...my intention was not to start my new pairs breeding inside the aviary, but they did it on their own of course, i was young and didn't know any better. So with 3 budgie breeding pairs I put in 6 nest boxes and twice as many food and water dishes, so each pair could have multiple box choices, and so each bird could have multiple food and water choices...everything was fine for a while, only 2 of the English budgie pairs laid clutches, one had 4 babies in their box and the other had 6...one morning I went to check on the babies and feed the budgies, and i can't begin to describe the horror to you...the nest box with 4 babies in it had the youngest 2 babies literally ripped apart, one was decapitated and the other ripped open...and their mother was dead on the floor underneath the box, also ripped open...in the other box, all 6 babies were fine, but the mother laid dead beside/on top of them, she was ripped open, she died defending her babies, as did the other mother, as the male who hadn't laid any clutches was covered with blood but unharmed, so he went after and killed both mothers and 2 babies, and then i was left with 8 babies and NO NESTING MOTHERS AT ALL, MEANING I HAD TO HAND RAISE THEM, AND ONE WAS ONLY 2 DAYS OLD...They all lived, but i was exhausted from feedings every 2 hours for the first 4 weeks...
So to answer your question, NO!!!! One breeding pair of birds per cage only!!!!