Neon and cardinal tetras school, "swimming all together most of the time", the other tetras swim in the same part of the tank, not necessarily in a traditional school. The recommendations for the # to have together can be because unless there are enough they will pick on each other. They should get along, unless like COBettaCouple said if the new arrivals are smaller, or if someone is sick or for some other reason being picked on.
One of the tried and true solutions to create tank harmony is to break up the line of sight for the fish. Give plenty of decorations, plants, rocks
driftwood, anything that will allow the fish hiding places until they get used to their new surroundings.