It appears the driver was on the wrong route to Ferrymead when he struck the bridge.
Witnesses said a resident on the street and pupils on the bus called out to warn the driver.
Woolston School principal Jeneane Reid said: "Some of the kids noticed and said 'We won't fit under there'."
Sergeant Richard Ryan said charges were likely but the police needed to finish their investigation and would consider proceeding under their alternative resolution system.
This involves talking to victims - in yesterday's case the children and adults on the bus.
He said this would involve consulting on the bus to see whether a resolution could be found other than laying charges.
Ms Reid said the school was "recovering" from the crash. Distressed pupils had been "pampered with hot chocolates" in the staff room.
Victim Support spoke to pupils and parents yesterday morning and the Ministry of Education's traumatic incident team was providing ongoing support to help the school identify which students might need counselling, said Ms Reid.
Red Bus chief executive Paul McNoe said the company did not see any need to stand the driver down during their own independent investigation into the incident.
He said the employee had been with the company for three months and was comfortable with driving after the accident.
Mr McNoe said he didn't know why the driver took that particular route, but the investigation would determine how events unfolded.
"We do our level best to make drivers aware of hazards in those specific areas that they are likely to go into.
"I don't believe that particular bridge is on that list, simply because it's not a normal route we would recommend."
Mr McNoe said: "My understanding is he actually slowed right down prior to hitting the bridge. If he had been travelling at 50km it would have been a much more serious crash with serious injuries, which it wasn't."
Woolston resident Amy Stoop said the noise was huge when the top of the bus hit the bridge a few houses down from her home.
She said a neighbour was sweeping up glass from a minor accident earlier in the morning when the bus came along the terrace.
"She told him to slow down because she knew that he wouldn't fit. But he didn't slow down, he just kept going."
When she came out to see what had happened Ms Stoop found glass and plastic all over the road.
The front windscreen and one side window had popped out on impact.
"There was a bit of a mess, glass everywhere. Kids were just crying, a bit dazed and confused. I just felt so sorry for the bus driver, but silly guy."
Ms Stoop said her neighbour who tried to warn the driver thought he was going too fast considering the bridge he was approaching, the size of the bus and the age of his passengers.
"He should've stopped and then analysed the bridge before he attempted to go through it, but he just went straight on."
She was surprised a bus would try to drive under the bridge, which she said had often caused issues for other vehicles.
"Mr Whippy lost his icecream, we've had people lose bikes, so it's a deceiving kind of a bridge," she said.
"There isn't really much signage until you get there."
Sergeant Richard Ryan said the children were more shocked than hurt.
"A few that came off [the bus] were quite visibly upset."
Ms Reid said a teacher and the pupils sang their school song, Tenei te Ruru, to keep calm.
When the shock wore off, many were disappointed about missing out on the zones cross country event she said.
"One of the children in the hospital was certainly complaining to the doctor that he'd have to wait another whole year."
