To answer my own question then:
I added three new tetras, identical in colouring to the original three tetras. The additions were smaller than the existing ones.
Within two days, all three of the new additions were dead. I am not sure which of the other community fish (guppies, albino catfish, angel fish) were the main culprits. I did see the original neons occasionally bother the new arrivals, as did the guppies.
A complicating factor is that I had, earlier in the day, moved the aquarium from one room to another. I waited over six hours after moving the tank to allow the fish to settle down. All the original water was retained and returned to the tank after the move. I had assumed that leaving the tank for a substantial period would have allowed the fish to recover before adding new fish. Perhaps I was wrong.
I had also added two fancy-tail guppies to the tank at the same time as the three neons. These are doing fine.
EDIT: I added three more neon tetras, which I chose to be as close as possible to the size of the original three. One month later, these new additions continue to do fine.