In general Google has pledged to remove H264 support in Chrome so, it's not really to be trusted.
Anyway...
That test page doesn't work in my Chrome Version 24.0.1312.57 m on Windows 7 - it only plays the audio and a blank video stream.
I did a quick survey and of the 22 people in this team that tested, 7 could not play the video and 15 could. Same versions of Chrome mostly, some on development channel with version 26, same Windows patch level and even some using the same video cards and drivers getting different results.
The videos tested are mp4 containers with one video stream in H264 format (not mpeg-4).
I've been unable to identify the cause but it's starting to become a frustrating issue as for all those that had problems with playback, no popular HTML5 fallback solution works. Even the homepage example video for things like ME.js, videojs, jwplayer and so one fails to play back video in Chrome.
Either Chrome has some issues with H264 support, or there's an unknown bug in some other software we all use... I'm going to do testing outside of the company network to see if there's a content filter messing things up.