Revealing mistake: After the water tentacle sequence, Coffee brings Schoenick back to their bunks to get the submachine gun, so they can take over the rig. The bag containing the gun is stashed behind some pipes up by the ceiling. Watch when he reaches behind the pipes to grab the bag--the pipes move at least six inches. (01:31:57)
Revealing mistake: When the crane falls down in front of the rig and then over the cliff into the abyss it's as if the rig was already just a few yards away from the abyss but when the umbilical pulls the rig again (after Bud says to grab onto something) suddenly there's enough space for the rig to be dragged at least a few hundred yards as if the rig never moved from its touchdown position earlier in the film.
Revealing mistake: Every time the aliens appear anything electrical shuts down but somehow Lindsay's camera is able to take pictures of the small NTI before the electricity powers back on.
Revealing mistake: When Bud is unscrewing the trigger from the nuke down at the bottom of the abyss, there are no threads on the mechanism to require unscrewing. The part of the device that would have the threads is plainly visible.
Revealing mistake: When he's on his way to send the nuke down the abyss, Coffee leaves the other SEALs to guard the rig's crew in the kitchen. He goes through a door, locks it and secures it with his belt, and then bangs on it to test its security. It moves more than a metal door in a metal housing would. (01:37:43)
Answer: Most of it was television news coverage about the increasing tensions between the US and USSR as it was playing out earlier in the movie. Additional archival footage from previous years' wars implied that the NTIs had been on Earth for quite a long time.