Frustrated customers wrote on the bank's Facebook page about the problems they'd experienced.
One said her she was embarrassed she couldn't pay for her groceries at the supermarket and books for study.
Another said she couldn't pay for prescription medication.
"I have transferred money from our BNZ accounts into our main transaction account so I can pick up medication requirements from the pharmacy. They money has gone from our account," she wrote.
After customers wrote they were disappointed there were no updates on social media, BNZ responded: "Hi all, thanks for your feedback, we are trying our best. Appreciate your frustration, and we share it - it's never a nice thing to hear you've disappointed your customers. Please let me know if there's anything else I can help you with. Thanks."
The BNZ spokeswoman was not able to say how many customers were affected.