You may be a tax resident in both New Zealand and another country or territory. In this case, DTAs have 'tie-breaker' rules. These determine which country or territory you're considered a tax resident of for the purposes of applying the treaty. You are still a tax resident in each country for other purposes.
For companies, you may need to apply to a competent authority to determine which country or territory you are resident in. Failure to get a determination can result in the loss of benefits under a DTA.