Searchers are continuing to scour rugged terrain and treacherous seas of an Auckland beach for the 21-year-old.
Ms Bambus told her friends she was going for a run at Piha on Friday morning, but never returned.
At a press conference on Sunday, a police spokesman said people have survived longer than this but as time goes on they have greater concern for her welfare.
"We very much remain hopeful that she remains alive.
"We really don't know what's happened to Kim ... All we know is that she's missing and we're trying to find her.
"It's an open mind. There are an endless number of possibilities because at this stage we don't know what's happened to Kim."
Police and search and rescue volunteers spent yesterday combing the area, with teams abseiling down the sheer cliffs to look for any sign of Bambus. Helicopters also scanned the coastline, while Bambus' family and friends anxiously waited at the track's entrance hoping for news.
Her sister, Storm Bambus, said she got a call on Friday night from one of Kim's close friends saying that she had not returned to her Ponsonby flat.
"I drove down and obviously the search and rescue started. There were helicopters all [Friday] night, dogs searching for her," she said. "We're assuming that she went for a run or a walk there and she hasn't been heard of since."
Kim had just started working as a registered nurse at Middlemore Hospital a couple of months ago.
"She's awesome," Storm said. "I couldn't ask for a better sister."
The family is from the Bay of Islands but Kim moved to Auckland about four years ago.
Kim's parents were on their boat off Great Barrier Island when they received news of Kim's disappearance and sailed through the night to get back to Auckland.
Storm said she last spoke to her sister mid-week, and they chatted for about 45 minutes.
"Pretty much it's just a waiting game now. We're hoping now that the media's involved that people will get in touch with the police with any information - if they saw her walking or anything - so that we can build a timeline and narrow the search area or get an idea of where she was or wasn't. That's what we really need now."
Kim Bambus was wearing a dark jacket, pink exercise top, black shorts and black and white Nike running shoes. Her hair was tied up in a ponytail and she was carrying a large water bottle.
She was last seen in Countdown in Williamson Ave, Ponsonby, buying snacks at 9am on Friday morning.
Land Search and Rescue spokesman Alan Wilkins hoped Ms Bambus was safe and well, saying the biggest threat to her could be the weather turning bad.
Police are asking anyone who may have seen Kim Bambus to contact Auckland Police on (09) 302-6400.
