It is possible that the old SIM is to old for the new device. It happens. It is also possible that the old SIM is from a different network than the new phone runs on.
If you got the new phone from a different company than the old SIM is through, then you need to check if they are in fact running on the same network (CDMA or GSM). If they are different, then they won't be able to work together. If, however, you go a phone through the same company that your SIM is through, then you need to contact them to get a new SIM.