The roof of the Home Building Centre on Old Airport Road collapsed on Kevin Olson, 24, and Cyril Fyfe, 41, while they were inside the shed fighting a 7 a.m. fire.
The investigation into the fire is not criminal, RCMP spokesperson Kerri Riehl said.
"It could be concluded today; it could take another week," said Riehl during a March 18 press conference.
Television news footage shows that four firefighters - trying to cut a hole in the roof to fight the fire from above - also fell on top of the men.
Olson, originally from Calgary and new to Yellowknife, was pronounced dead shortly after being pulled from the building.
As of 2 p.m. on Saturday, Fyfe was on life support and reported to be in critical condition at Stanton Territorial Hospital, though the nature of his injuries was not released.
RCMP, the fire marshall, and the Workers' Compensation Board are doing the investigation.
Fire Chief Mick Beauchamp said he would not talk about the competency of the firefighters until the investigation was complete.