Improved technology enabled specialists to treat babies born as early as 23 weeks, who would have to stay in hospital four to five months.
"We can perform great miracles, but we can't speed up time," Ms Escott said.
The units have a website on which they post updates on the number of spare beds.
"The website has been 'no beds' every day," Ms Escott said.
"New Zealand is a country of four million people and, for goodness' sake, we need to get ourselves organised and discuss it."
Neonatal Trust chief executive Michael Meads said there was no crisis yet, but action would be needed.
"As our population grows, so do premature births, or at-risk births. It will only get worse and there will only be more demands on hospitals to increase their capacity."
A first-time mother has told how the Wellington unit saved her son, who was born 24 weeks into gestation - 16 weeks before his due date.
Lachlan Bain arrived on November 10 weighing 880g - less than an average bag of sugar.
His mother, Michelle, was flown to Wellington on an emergency flight from Nelson after she went into labour the night before.
"[Doctors] have a scale of one to 10 that rates signs of life, and when he came out, he was two. There was pretty much no sign of life," Ms Bain said.
She has spent the past nine weeks at the neonatal unit, watching every gram of weight Lachlan has gained and despairing over every setback.
Lachlan was born with respiratory problems, and a valve in his heart remains open.
The worst-case scenario would be an operation to close it, Ms Bain said.
At the hospital, she and her husband, Don, have met numerous families in similar positions.
One couple were sent to Wellington because there were no beds at Starship and Middlemore hospitals in Auckland or at Waikato Hospital.
The Ministry of Health's chief adviser on child and youth health, Pat Tuohy, said the neonatal intensive care units were designed to run close to capacity most of the time to maximise efficiency.
Dr Tuohy also rejected claims that a growing population would place more pressure on the network.
The number of births in New Zealand had decreased between 2008 and 2011, he said, and the per-centage of premature births had remained "fairly static" over the same period.
Dr Tuohy said a new unit being developed at Dunedin Hospital would have more flexibility in its capacity than the current one and was expected to be open at the end of the year.
Meanwhile, Lachlan is 9 weeks old today, but he could be calling Wellington Hospital home for a little longer yet.
"I'm hoping to leave in a couple of weeks," Mrs Bain said, "but one of the first things you learn at neonatal is that you lose control over everything."
Premature births
• A normal pregnancy term is about 40 weeks. A premature baby is defined as being born before 37 weeks.
• The youngest age at which babies can be looked after is 23 weeks.
• The cost of equipment to look after one baby is about $500,000.
• The smallest baby cared for at Wellington Hospital weighed 400g at birth - less than a block of butter.
