When the iPad won't play an MP4 video

I had trouble serving an MP4 video to my iPad. Although the video complied with all the specs (H.264 codec, up to 1080p, 30 FPS) I always got this error:

This video could not be loaded, either because the server or network failed or because the format is not supported: http://10.40.0.177:3000/video.mp4

After spending a lot of time fighting this issue, I tried to upload the same unchanged video file to a production server (Apache / Passenger). It worked immediately. I guess iOS is picky about some HTTP header that my local HTTP server (Thin) didn't provide.