Continuity mistake: In one of the scenes Hopkins is trying to make what I think is a fish hook or something. He's obviously been in the wilderness for a while so he's not clean shaven but he isn't exactly beardy either. However, during the scene with this fish hook his beard changes from moderate to bushy to moderate again within seconds.
Factual error: Charles and Robert manage to outrun a charging Alaskan brown bear over a course of several hundred meters, leading it into a trap. This is absurd. The absolute maximum running speed of a human being is about 27 kilometers an hour, and that is for an appropriately dressed, fit athlete over a very short course on flat ground. They are in a rock strewn stream and are wearing heavy winter clothes. A fit, healthy, active male Alaskan brown bear like the one chasing them could hit 45 kmh in that environment without popping a sweat and could keep that up for a kilometer or more. When enraged or charging prey - as this one was - they have been clocked at 56kmh. This is nothing to do with an adrenaline rush - that will not enable Charles or Robert to exceed their body's maximum running speed by 200%.
Continuity mistake: When Charles is showing Steve the constellations, the sky is pitch black. Yet in the next scene when Charles goes down to the shore to Bob, the sky has brightened considerably. No indication is given that a good deal of time has passed between these two scenes.
Revealing mistake: When the needle is set in the pool of water in a stump to act as a compass the pool has been carved out with a chain saw very recently and clumps of the wrong moss set around the rim like it belonged there.
Other mistake: I've always wondered where they got the wood to make the enormous fire on the top of the mountain. You can clearly see that there are no trees anyway nearby, (except for several hundred meters further down) and they are not carrying wood with them either.
Visible crew/equipment: While the 3 men are using a tree to cross a gorge during their first encounter with the bear, you can see a black metal box with white writing on it, sitting on a rock formation in the background. Definitely filming equipment or something. You can get a glimpse of it directly behind Charles as the 3 of them sit and gather their breath once the action breaks.