Their last opportunity saw them lose the Oceania Cup final to Australia.
Hockey South Africa says the "crippling" decision from their Olympic Committee to stop its teams from going to Rio and will have a long lasting negative impact on the sport in South Africa.
Hockey South Africa has apologised to the athletes.
Had this decision not been made, the New Zealand men faced temporary sporting oblivion. A loss of personnel, many of whom balance jobs with international careers, seemed inevitable.
"Our athletes train full-time while incorporating work and family life, which places tough demands on them," HNZ high performance manager Terry Evans said last month.
"We effectively do that on a shoe-string, smell-of-an-oily-rag [budget]."
Hockey NZ received funding of $750,000 for the men's team and $1.3 million for the women's team in this year's High Performance Sport New Zealand budget.
The women are guaranteed another $1.3 million next year. There are no such promises for a men's side who last missed the Games in 2000. Decisions from High Performance Sport New Zealand on next year's funding are due before Christmas.
The women are on discretionary performance enhancement grants but need to get a top-three finish in December's World League 4 to guarantee that funding in 2016. The men are already out of contention.
"We need to find a mechanism by which our athletes can commit," Evans said. "In the current environment that's difficult, because we're working to make the whole operation sustainable.
HPSNZ chief executive Alex Baumann acknowledged the desire to support hockey as a key part of New Zealand's sporting fabric, but says sports still need to meet agreed standards.
"Team sports can impact a broad range of people and therefore be important for the country, but we can't shy away from actual performance. We understand team sports cost more and our hockey teams probably receive less investment relative to some of the stronger countries.
"However, as our system has evolved, the bar has been raised. We would love to invest more but this is the reality."