The qualification process is convoluted. The top three teams from each of two world league tournaments - the second starting in Antwerp, Belgium next week - progress to Rio, as do the winners of the five confederations, Europe, Africa, Asia, Oceania and Pan America.
However there will be double-ups. For example, Germany or the Netherlands will be expected to win the European event, and qualify through the world league; as will Australia, if they beat New Zealand in October's Oceania tournament in Taranaki.
So there is wriggle room and an optimism among the New Zealand camp that they will still make the Olympics. But yesterday's defeat was a heavy blow and leaves nerves jangled.
"We should have won the game in normal time," experienced midfielder Shea McAleese said.
Canada were tough and resilient but spent the final quarter in a major defensive operation.
In one period during the shootout, which took a stunning 28 attempts to find a winner, eight successive penalties were saved by New Zealand's Devon Manchester and his Canadian counterpart David Carter, with some spectacular work.
Veteran Ryan Archibald missed all three attempts; captain Simon Child missed both his. By contrast, midfielder Steve Edwards converted all three of his chances impressively.
New Zealand did have some bad luck.
Archibald's first attempt crossed the goal line a fraction after the eight-second hooter sounded.
Blair Hilton's second deflected off the cross bar.
In a game of centimetres, either of those would have had New Zealand through.
"Devon was absolutely phenomenal in the shootout and did all he could to help us win," McAleese said of the dramatic finish.
"It was crazy to watch.
"You went through the ebbs and flows of emotions and by the end of it you were almost feeling sick in terms of the emotional roller coaster," he said.
McAleese said it was up to the Black Sticks to regroup for a game they could not lose.
They beat 16th-ranked Japan 2-1 in pool play this week.
Finishing fifth, by their calculations, gives them a strong chance of Olympic qualification once the double-up qualifiers have been sorted out.
It's not the way they wanted to have to get to Rio, but at least they still have a chance.
