The steel would be designed with safety factors. Every single piece would be able to hold 1.2 times the Dead load (self weight, equipment weight, i.e. weight you specifically know will be there) PLUS 1.6 times the Live Load (people weight, non fixed fixtures, desks, paper, so on and so forth) without yielding.
But if the steel starts losing strength, it'll suddenly start yielding. If it starts to yield and the weight isn't taken off a little bit, it'll fracture suddenly.
So as the fire was burning hotter and hotter, it was turning the steel into string and the concrete was being burned through. The main problem with the concrete being burned through was the reinforcement steel. Without it, the concrete can only hold weight in exactly one direction.
That's not forgetting buckling. All the columns that weren't on fire wouldn't be able to hold all that load, which is what I suspect (i'd have to check) is also why the building collapsed suddenly.
And one more thing. Fireproofing isn't really fireproof. All it does is delay the heat transfer. A good layer of fireproof would take 2 hours to burn through. This means the occupants have 2 more hours to get out safely (give or take). However, the fireproofing was all blown off of the columns when the planes hit. Thus the steel was instantly exposed to the fire.
