Cycling does indeed have to do with it, as I'll explain.
Shoaling fish like all tetra need to be in a group, as you clearly know. Part of the reason for this is the hierarchy the fish will establish within the group. This manifests itself differently depending upon the species, but it is an inherent need of the species. When the fish are settled, and free of stress, there will usually be no significant physical damage with most species [some are different, like Serpae Tetra who will fin nip regardless]. But when the fish is under stress, this changes, and usually for the worst.
Ammonia or nitrite above zero is very stressful. The higher either level is, the more stress. There is also internal damage being done, and this adds more stress. Keeping the fish in too small a group can cause this too, as can too small a tank space, just so you know and to complete the picture.
Fish have few ways of dealing with stress. Sometimes they become weak and waste away, refusing to eat. But more often they take the opposite approach, and increase aggression. It is their only way of fighting back their frustration.
I certainly wouldn't add more fish until this settles down. Meanwhile, make sure ammonia and nitrite are at zero by doing daily water changes using a conditioner that detoxifies both, such as Prime or Ultimate. Are there live plants? Not only do neons and most tetra need the shelter of plants, and floating plants especially; these also help with the cycling.
Byron.
