For some reason, I get this right all the time but I can't remember understanding how the puzzle is solved.

The bug is that the 'animation' of the bridge doesn't work right. Sometimes it shows as being down as soon as you enter the room (even though it's really up, you fall through it when you try to cross it). Sometimes when you get the bridge down it will immediately jump back to the up position again (even though it's really down). The trick is to listen to your companions, when you get he right combination on the levers and the bridge is really down, one of them will say something like "You did it!"

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Should also work. There are multiple combinations that will get you there. Usually I just pull some levers at random and within a minute the giant troll nord will start cheering that I did it.

Thank you! Took me about 10 minutes of making a correct combination and messing it up before it worked. Looks like some sort of bug.
Also I could walk through those small metal doors without opening them for some reason. o.O