Once you click one of the video links, it will open up in LW & start downloading.
hmm ... considering that is a site, not another user, then it does sound suspicious. Those magnet link files are free to copy.

A couple of options I can think of. (1a) One is back up all your settings in LW, make a note of them all. Then delete the LW prefs folder whilst LW is closed; instructions here: Point #2 & 3 in Fixes for Limewire (sample image included at bottom of page.) Vista users see http://www.gnutellaforums.com/genera...der-vista.html

(1b) Make sure your days to keep incomplete files is set to a large number such as 9999 days. The default age is 7 days.
(1c) Make sure your hard disk drives have plenty of space on them. And Defragment the drives occasionally. Remember, you can always right-click a download & change location or choose save as & choose location, if your main drive or partition is getting too full. In that case, I'd suggest changing your save/share location.

(2) Try another gnutella program such as Phex & see if same thing happens with those magnet link video files: Phex - Download
You can always uninstall Phex later. But if same problems with Phex, then sounds like your ISP's are causing the problems. BTW Phex has some nice attributes. One being that it connects to more gnutella client programs, unlike LW which mostly only connects to other LW users. And Phex has a size filter. Phex's downside is the interface is not as intuitive to new users as LW is. It takes a little to get used to understanding how it works. (Note: for magnet links to download in Phex or another gnutella program, you will need to change default magnet handler program. Otherwise LW will try to download it as it does now.)