People trying to contact Immigration New Zealand are waiting on average 44 minutes for their calls to be answered, as the department tries to cope with "unprecedented" demand for its services.
Immigration New Zealand (INZ) said it was receiving 60 percent more calls than it could handle and it was struggling to find staff because of low unemployment.
The average waiting time over the last month was 44 minutes, a spokesperson said.
But one couple has described waiting two days to get through to a phone operator.
John and his British wife, Lillian*, who is a permanent resident in New Zealand, needed to get immigration information for superannuation applications this week.
They tried calling the INZ customer service line on Monday, after failing to find the information online.
But they were placed at number 300 in the queue. They then tried calling again ... and again. This time they were able to reach the call-back service.
"Someone called back after several hours, then the line was a bit crackly and so the guy said, 'Look I'm sorry, I can't hear you, well,' and hung up."
After being hung up on, they did not hear back, John said.
So the pair went through the whole process again the next day and again on Wednesday morning. Finally, that afternoon, they heard back.
"They do call you back but it's a rigmarole getting through."
It would be doubly frustrating if English was not your first language or if you had limited access to the internet, he said.
"To make those people wait a long time just to be able to leave your number and then the person might ring you back, it's not a very good service.
"Especially if we want to encourage people to New Zealand. I'd imagine a lot of people just give up and don't bother."
INZ admits that customers have endured unnecessarily long wait times on the phone, due to an "unprecedented demand fuelled by an increase in people wanting to come to New Zealand".
In a statement, it said the contact centre received about 1.6 million contacts via phone and email per year.
"The contact centre is receiving 60 percent more calls than it can handle at present."
About a third of enquiries were about the status of visa applications, it said.
It said although the wait time to get through to an operator was long, it offered a call-back service so customers could maintain their place in the queue without needing to wait on the line.
"This unprecedented demand has also caused the telephone system used by the contact centre to become overloaded."
The call centres, in Auckland and Palmerston North, were attempting to employ more staff.
"[But] recruitment has been slower than expected," the statement said.
"We believe this is because of low unemployment which has led to a shortage of suitable candidates."
It was working through the issues as swiftly as possible, including improving self-service tools.
*They did not want their surnames used.