Social Development Minister Anne Tolley acknowledged problems, but said they had been resolved after meetings with agency chiefs and some altering of contracts with NGOs.
Children's teams are a new model that aims to help families with serious needs but who don't need immediate legal action.
A multi-agency panel meets weekly to consider each referral and choose the best "lead professional" to work with a new family.
These professionals come from agencies - including NGOs and health, education, justice and social services - that get no extra funding to take referrals, and the team works within existing funding constraints to arrange a plan for the family.
Staffing problems meant the first four children's teams weren't able to help as many vulnerable children as desired or expected, documents released to Labour under the Official Information Act show.
"All of the current four teams are hampered in reaching the expected volumes of referrals due to the lack of lead professionals," a briefing to Ms Tolley on May 14 states.
Teams in Rotorua and Whangarei were only able to accept one referral in a month.
The Ministries of Health, Education and the Waikato District Health Board resisted the provision of lead professionals upfront, instead wanting to provide them on a case-by-case basis.
Ms Ardern said she had heard repeated, recent concerns from NGOs that they were being asked to provide staff, without extra funding: "It would mean pulling staff from existing families. It is robbing Peter to pay Paul".
Ms Tolley said that would not happen as agencies were already working with the same people that would be referred to children's teams.
The "bottleneck" had cleared after agencies and NGOs better understood the model, she said.
To manage the estimated 20,460 children who will eventually use children's teams, 284 lead professionals will be needed. At the end of June, there were 51.
Ms Tolley said it was her understanding teams now had enough lead professionals, and she was confident numbers could increase as needed.
"As long as they are releasing a number of families. It is something I am tracking very closely at the moment, which is why I have slowed the process down, to make sure we get it right."
Ms Tolley's strong preference was to have all 10 children's teams taking referrals by the end of the year, however the Clendon/Manurewa/Papakura team's opening has been postponed until next year.
Five teams are currently operating, with Tairawhiti (Gisborne) opening last Friday and Christchurch, Whakatane and Whanganui opening later this year.
