Revealing mistake: In the beginning, there is a scene where Sen. Shaw is watching Raymond make a speech on a TV screen. It was obviously put in in post-production because the aide immediately next to the screen is nodding and clapping in agreement at inappropiate times.
Revealing mistake: When Raymond Shaw's skull is being drilled to implant the chip, bone "dust" comes out. This never happens, since always some blood comes out and the bone debris are mixed with blood and therefore unable to "fly" like wood or plaster debris.
Revealing mistake: Sen. Jordan is supposed to be an expert kayaker, but when paddling his kayak it is very obvious from the way he splashes the tips in the water that he cannot paddle.
Chosen answer: Perhaps the piece-of-plastic killing was a "test run" by the brainwashers to see if Raymond would actually react to their commands. You can see other soldiers lying down wearing "halos", hinting that this happened at a different time and place than the observation room choking.