An astute Reddit user has discovered something strange while autopsying XCOM: Enemy Unknown. It seems that the game files include 1080p and 720p versions of all the cinematics; however, at least for some users, only the 720p versions will play, regardless of monitor resolution and video settings. Currently the only way to force the game to use the 1080p versions is to move the 720 versions and rename the 1080 ones to the 720 file names.
Luckily, you shouldn't have to do so manually. The intrepid Redditor, who goes by the dignified handle "loinbread," has released a batch file that will preform all the heavy lifting for you. (Remember to practice safe downloading. We didn't have any issues with the file, but we can't be held responsible if it turns out to be some sort of alien conspiracy computer virus.) Extract it to Steam/steamapps/common/XCom-Enemy-Unknown/XComGame/Movies and run it, and it will rename all of the 1080p files while dumping the 720p ones into a backup folder. The low-res videos (which are essentially just 4GB of dead weight) can be kept or destroyed at your leisure.
Now enjoy your high-resolution alien viscera splatter.