You didn't say what phone you have, but it, apparently, is not capable of receiving the "enhanced" messages (those longer than 160 characters). The balance of the message gets cut off.
Call CS and see if there is a way then CAN enable enhanced messaging on the phone you have. If that's not possible, then the solution is to have your friends send shorter messages, or multiple messages.
I should be able to get them though, because some longer ones do come through. I have a Samsung Reality, but even when I had my old enV 2 I could get long text messages. I thought Verizon phones could all get messages longer than 160 characters? It has stopped doing that for the moment though so hopefully it will continue to receive them properly (fingers crossed).
Were the senders using a signature? Possibly adding an emoticon or other special character?