The Copyright Tribunal is for the first time going after three people accused of music piracy.
The tribunal is in the process of identifying three internet users who have allegedly infringed copyright through file-sharing.
The Telecom customers have each incurred three warnings under the controversial "three strikes" copyright law. The maximum fine for the offence is $15,000.
The Recording Industry Association, which represents big music labels, initiated the action - the first time it has gone this far despite almost 3000 warning notices being issued since the law came into effect a year ago.
Telecom says it can be hard proving just who in the household infringed copyright - for instance, if the account holder is a parent but a child is doing the downloading.
The Ministry of Justice says the next step is to obtain account holder details from the Internet Service Provider to identify the suspects.
Internet New Zealand says the ISP is the only one who can identify what computers illegally downloaded the material and its account holder - but the difficulty is establishing who used the computers.
It believes punishing the three customers for allegedly pirating music will not stop people from downloading illegal material. Spokesperson Susan Chalmers says the move might see an initial dip in illegal file sharing, but that won't last long.
However, an organisation representing more than 10,000 New Zealand artists says it supports the action taken.
A spokesperson for the Creative Freedom Foundation says it is good the law is being used to protect artists' work. Bronwyn Holloway-Smith says it is important that their work is respected, not abused.