Another customer queried the safety of her funds after receiving an email that appeared to be a hoax.
"I am a long time BNZ customer and this morning I have received an email supposedly from you telling me my bank account is seriously overdrawn.
"I immediately treated this as a hoax but you need to know as clearly you have been hacked! That being the case can you guarantee your customers funds are safe????" said a Facebook post by Jan McKenzie-Laurence.
A spokeswoman for the BNZ said a small number of customers had received routine text an email alerts for accounts that did not belong to them because of a system error.
"These alerts did not include customer account numbers or names."
She said the bank was still investigating the cause of the issue and apologised for the inconvenience.