I never noticed this before, and someone just made me aware. If you start at around 3:00, when they start improvising around Frere Jacques, and watch to the end, you notice that, not only is Picard's flute dubbed in from someone who knows how to play, but he''s got someone else's hands holding the flute from behind him.
[youtube]http://www.youtube.com/watch?v=p3FgJztHqFE[/youtube]