Students are demanding the government get rid of the limit on loans, which they say unfairly affects medical students who already have undergraduate degrees.
The government introduced a seven-year borrowing limit in 2010, to encourage students to finish their degrees quickly.
It was extended to eight years in 2015 because the government said more students were entering medical studies with prior degrees.
It said it expected the number of students affected would grow from 10 to about 100 each year by 2018.
But a spokesperson for a campaign to end the cap, Kera Sherwood-O'Regan, said there were now at least 140 students who faced having to complete the final years of their medical degree without the assistance of a loan.
Third-year medical student Freeman Apou pursued undergraduate degrees in genetics and English to boost his grades after he initially missed out on acceptance to medical school.
Four years later, he started medical school but was now faced with finding $60,000 to get through the last three years.
"[It] absolutely gets overwhelming sometimes," Mr Apou said.
"I could be doing all of this for nothing. All this hard work trying to get an 'A' and trying to pay rent at the same time, as well as save, and if I don't have that money, I've absolutely done it for nothing."
The cap particularly disadvantaged medical students who did not have wealthy enough parents to provide a safety net, Mr Apou said.
"As a society we view doctors as this elite class of people who come from wealthy backgrounds - and absolutely there are those particular types of students in the class whose parents could fork out that kind of money - [but] for a great part of the class that's not true."
Ms Sherwood O'Regan said a six-year medical degree was the longest in the country and the most expensive.
She would have to find $75,000 dollars to complete her studies, she said.
The students were not looking for hand-outs, just longer loans, she said.
"Half a doctor isn't very useful for the government and it's not very useful for taxpayers.
"So really [the government is] essentially undercutting their own investment by not just simply removing the cap for those students just to let us finish so we can go and be qualified and work in the communities that we're intended to serve in the first place."
'We're not going to have Māori people there'
Māori Medical Student Association Te Oranga tumuaki Chayce Glass said 18 out of 300 Māori medical students would not be able to graduate due to the cap.
Māori only made up five percent of the medical work force in New Zealand but 15 percent of the population, he said.
"What I'm worried about is we're not going to have Māori people there to provide health service delivery to our own ... people."
While there were scholarships for Māori, many did not cover the costs and were highly competitive, Mr Glass said.
"What I don't want to see is whānau being overburdened by trying to find $25,000 per year just to fund their children to be able to finish their medical studies," he said.
The student medical associations have more than 3000 signatures on their online petition and said they would take it to parliament if Mr Goldsmith refused to meet with them.
Tertiary education minister Paul Goldsmith said he would meet with medical students to talk about the effect of a time limits on them.
He said the duration had already been extended to allow students to claim the loan for longer and the cap was in place so students could complete their studies as quickly as possible.