This works on a Lenovo T60 1952 APY laptop with 3.28 Gigs of desktop memory (4 Gigs intalled - 32 bit system) running Linux Mint 13 LTS Mate edition.
This doesn't work with all video sites. The encryption on Flash video is getting better so this trick will work most of the time but not 100% of the time. Start the video and allow it to play. If there's a download indicator that moves ahead of the time mark then let the video play through far enough so the entire video is downloaded, then pause it.
Open your home folder and press Ctrl-H to show your hidden files. Click on .cache > chromium > Default > Cache, inside that folder you'll see a video icon with a thumbnail of the first frame of the video. Copy and paste the file to wherever you want to keep it, then rename the file, the video is now yours.