Neons develop a pecking order, since they are new this is probably what they are doing. Mine did this for the first few weeks and now they are content chilling. The two that died may have not acclimated well to the new conditions as Neons tend to be a little on the delicate side. If this was the case I would suggest doing the "drip acclimation" method. But, then again, maybe you just got one Neon that's an a**hole.
i agree wit hthis 100%. also i was wondering if you had like all males and 1 "eggy" female. the 1 male to 2-3 females rule applies to livebearers, but i am starting to think that this may be the case in some egg layers. i had a family member who had the same problem and when he sexed the "random" lot from the lfs he realized that they got mostly males and only like 1-2 females and the others were more aggressive. i myself have not experienced this so i don't know if it was just a fluke or not. i have had neons die from poor acclimation and even proper acclimation and just them being sensative to water conditions. i noticed that if the tank is less than like 6 months old they take forever to be "normal healthy fish". i am not saying that just because you're tank is new, that the neons will die. i have had success doing a full cycle with neons in the tank (i don't recommend this-what a headache!!!).
also from my experience... neons are nippy little bastards (sorry about profanity) and can never be pleased. i just stop getting them for this reason. they will nip at anything and cause chaos in your tank (even amongst themselves)
all of this may not be conventional and factual (relating to the species in general) but from my experiences (mostly bad) i don't deal with them much. i will get them if i have a lot of room to stock fish, but in a new tank with a bunch of other fish i will never do that again...