Did you try doing a Bitzi lookup on the file to see if any complaints or see if the files are listed at all? Just a thought it is file related.

The only time I've seen something similar is when LW detects a corruption or part of the file missing & attempts to redownload, but is perhaps struggling to redownload that part & have it match the download hash.

No, I didn't; still the problem doesn't seem to concern one particular file only but all large video downloading, TV series (350 Mo or so) or full-time movies (700 Mo or so). When LW detects a corruption we are immediately notified by a window opening warning the user that the downloading is going to stop.
This problem of downloading getting suddenly frozen while apparently going on has occurred only very recently.I shouldn't be the only person concerned.
I first thought it could be because my subscription to LWpro had run out but even after renewing it the problem persists.

Now it's working; I didn't do anything except that I sent a message to the LW support website. Did they do anything about this kind of problem,I don't know, the fact is that's working now