Beneficiaries calling Work and Income for help say they are enduring long waits - often of more than an hour - to talk to someone.
Work and Income comes under the Ministry of Social Development (MSD), which says it has had to re-direct staff to work on a Covid-19 support helpline.
One woman, who RNZ agreed not to name, has been ringing WINZ to get help with an accommodation supplement this week and has not been able to get through.
"When I tried to call them, all I got was a recorded message saying that they were too busy to answer my call right now and to try again later," she said.
When she rang last week for help because she had run out of food, she did get to speak to someone after waiting an hour and half on hold.
She said her friends had been having similar problems, either unable to get through or facing long waits.
When RNZ rang the helpline this week, the wait time was over an hour.
And for this beneficiary, visiting an MSD office is out of the question.
"Because I don't have transportation, I cannot get to a WINZ office," she said.
"My closest WINZ office would either be in Alexandra or Queenstown.
"There is no public transport to Alexandra, or I could catch a very expensive InterCity bus to Queenstown."
Even then, she said the buses to Queenstown were irregular and she would probably have to pay for accommodation overnight before she could return home.
"It's been incredibly stressful and really triggering for me as well."
MSD also runs StudyLink which takes care of students' loans and living costs.
One person told RNZ she had been trying to get through on that helpline for a week, but when she tested positive for Covid-19, it took the ministry just 24 hours to ring and offer support.
MSD group general manager for client service support George van Ooyen said he understood phone delays could be frustrating but it was an incredibly busy time for them.
"While some people are having trouble getting through, we are also helping large numbers of people each day," he said in a statement.
"Last week, we answered 96,372 calls through our call centres. The average wait time this week to contact Work and Income is just over 40 minutes."
Van Ooyen said the ministry also had to pitch in to help with the Covid-19 health response, with a welfare helpline for people who need support with supplies and money while self-isolating.
"The number of staff we have redeployed is between 400 and 500 per day, fewer on weekends," he said.
"All of our staff are working hard to look after everyone as quickly as possible. We have 10 contact centres with more than 1200 staff.
"There is also a call-back service for callers who provide a client number, pass an identity check and provide a phone number."
Another beneficiary RNZ spoke to said she had been having trouble getting through for a few months now.
"You'll ring and it'll say the waiting line is 90 minutes and to please call back later and then it hangs up," she said.
"It's been a big pain for me, because I missed a call which caused them to cancel my benefit.
"And because I couldn't get through, it was two weeks that I didn't have any income and I ended up going into the [MSD] office.
"So now when I need help, I just go straight to the office."
Van Ooyen said as well as the Covid-19 helpline, MSD had boosted its staff numbers working in the community, who offer whatever support individuals and whānau needed to isolate safely.
"This includes connecting them to services available during, and in the transition from, self-isolation," he said.