Question: The idea is download a video in parts (from different servers) but starting to play before video file is complete.
The issue is that VideoDisplay component doesn't read the video file if it is opened for writing and vice-versa: writing is impossible if VideoDisplay plays the video...

Answer: 
I know I may sound totally off-topic, but you should make sure you've properly considered video streaming thru Flash Media Server, Wowza or Red5 before you put that much effort in downloading a file in chunks. Just a thought..
What about having multiple parts of the video, so playing can begin as soon as first part gets downloaded. So the parts have to be concatenated somehow at client side.

(Additionally, if one's connection is insufficient, showing a loading symbol is fine.)