Are you sure the neck was broken? I doubt it.

The last time I lost a bird, he/she hadn't been acting the same for 3 days. I was right there and mid step from the perch to my finger she fell dead. I picked him/her up and his/her neck was so floppy I thought it was broken.

I have no idea why he/she died but thinking back. It was spring so I opened windows during the day, could have been a draft. OR, 5 days earlier I replaced my 20 year old Revere Ware kitchen pots and pans with new fangled 18/10 teflon stuff. I learned that teflon gives off fumes that is deadly to birds.

Teflon can do it. We can't use our George Foreman grill indoors because it makes one of our parakeets ill. It happened every single time we used it, and he was never ill otherwise. We don't keep the birds in the kitchen, either, although our house is not large. We don't even use the thing anymore, if it's not good for the bird, it's not good for us.

Two pairs of lovebirds, with nestboxes, shouldn't be kept in the same cage. It is possible one of the older ones killed it but my guess would be it was from either egg-binding, stress, or starvation (from the others not allowing it to eat). A nesting, bonded pair of hookbills can get very territorial and make life miserable, and sometimes deadly, for any others in the cage.

Is this a large walk-in aviary or a typical pet cage? For the safety of the single bird, I'd set him up in his own cage.

This particular cage is 40" high (62" with the legs included), 32" wide, and 21" deep. I think it's plenty big enough for 4 lovebirds, but the nesting issue probably changes the equation. I have always heard you should buy the biggest cage you can afford (and is appropriate for the type of bird), and I've always tried to do that.
As far as the neck being broken, after reading gammyt's post, maybe it wasn't but it surely looked it.
It makes me very sad to think that I wouldn't have noticed trouble or illness, but it's certainly possible.
Annz, if I remove the mate from the cage, is it safe to put it in the other (smaller) cage with 2 other lovebirds ? These 2 seem to only tolerate each other; I haven't noticed anything else from them. Maybe I have the wrong idea about lovebirds, but I would think a single bird would be very lonely by itself.
About the teflon: 7 years ago, the day I got my first bird,(a CAG), I was warned about the dangers of Teflon, and the pots & pans went out the door the same day. I have since used only stainless steel, and have no regrets since that's one less thing I have to be concerned about.
Lastly: I don't really want babies, but they seem determined to lay eggs. I can remove the nestboxes (I'm aware that they encourage breeding) but when I do, they'll lay eggs anyway, even if they have to do it in their food dishes. Incidently, the remaining bird is still sitting on the eggs.

So there are eggs in the 'birdie hut'?
You're right that nesting pairs change the equation within a cage. I'd be very hesitant to put a single bird in cage with 2 other bonded, compatable birds. I've seen, and heard of, 'pairs' that immediately killed the intruder.

Re: the egg laying........it's breeding season now, but are you feeding the birds a high protein diet year round? If you don't want them to breed, cut back on the protein , feed a basic diet and remove all nests. Have you candled any of the eggs to see if they're fertile?

You may be able to leave the current 3 birds together but I'd try to put food and water where the single bird has easy access to it. Just remember that if the two in the nestbox are a true pair the single guy will always be an outsider!

Lovebirds don't have to have another bird with them to be happy and they'll do fine by themselves with toys and the other birds nearby. When introducing one or a group of birds to another group, it's always best to put everyone into a totally new cage all at the same time. That way no one has established territory and it will help avoid fighting.
But........always keep a close eye on the birds to make sure everyone is getting along well.

my grandparents had 2 buggies and I'm not too sure if it was the female or the male that killed the other.

Yes, both pairs had 3 eggs each. The loner is now sitting the eggs it already had. For the time being, I have left him or her in the same cage, and provided separate food & water. Things seem to have calmed down a little, but I'm not dropping my guard yet. BTW, the loner is the grown offspring from the other pair. That pair had 3 babies; the one that died is also theirs, and the 3rd one is the one in a separate cage with one other lovebird that I purchased. Hope that's not too confusing ! I do like the idea of using a new cage when combining birds; I have the space, but not the $ right now.

I would move the lovebird into a cage by his self. The other 2 will undoubtably keep him from eating and he will die as well, it does not take very long at all for a bird to die when they cannot get to food. lovebirds are not really that lovey in my opinion, they can be very mean. Do not stick him in a cage with established lovebirds, they will do a lot more than just keep him from eating.
Even if this is a offspring they can possibly still kill it, my mother had 4 from the same clutch in togather, they all ganged up and killed 1, then 2 ganged up and killed another one, she got rid of ALL the lovebirds after that, another thing, nesting birds should always be seperated, it never works having more than one pair in togather.
If this was brother and sister laying these eggs, they are probably not viable, but I am not positive on that, my mother would know better but I am thinking I remember her saying something about that.