I haven't done much video work in flash before but I've never had issues importing video media...
I'm assuming your working with Flash MX, if your using flash 4 or flash 5 then I can't help you, I heard that earlier versions of flash had really poor video support. Also, make sure the publish settings for your movie aren't set to publish the .swf file as anything other than a flash mx file.
Does this happen with just that video file or all AVI files or does it happen with all Video files? Does the video have sound, if so can you hear the sound playback in flash?
If you can hear the sound playback but you can't see the video image, then it might be a codek issue. To see what codek a video uses right click on the file and select the 'properties' option, then click the 'summary' tab under windows XP/2000, look for what's specified next to the "Video compression" title. I have no idea how you'd view this information on a mac.
Try playing the file in windows media player, if you can't see the video there then you don't have the required codek installed.
If you can, try converting the video to a different format, like a QuickTime movie or an MPEG.
Also try creating a new blank flash movie and importing the video to that, just to see if the video file is getting lost in your movie somewhere.