The problem you are suffering is that not all browsers support all video formats and / or audio and video codecs, so if the android chrome browser does not support the one you use, it will not play correctly.
Solving the problem HTML5 makes it easier for you to work with multiple video sources:
<video poster="http://i.vimeocdn.com/video/578146756_1280x720.jpg" class="fullscreen-bg__video" controls width="560" height="360">
<source type="video/mp4" src="http://techslides.com/demos/sample-videos/small.mp4" />
<source type="video/webm" src="http://techslides.com/demos/sample-videos/small.webm" />
<source type="video/ogg" src="http://techslides.com/demos/sample-videos/small.ogv" />
<source type="video/3gp" src="http://techslides.com/demos/sample-videos/small.3gp" />
</video>
I recommend you have the video, at least, in MP4 and WebM.
Here is an example without selection that works for me in Chrome for Android:
<video poster="http://i.vimeocdn.com/video/578146756_1280x720.jpg" class="fullscreen-bg__video" controls width="560" height="360">
<source type="video/mp4" src="http://techslides.com/demos/sample-videos/small.mp4" />
</video>
However other MP4 videos do not work for me, so it's probably due to an unsupported video or audio codec, but having alternatives the browser is able to choose the one that I can use correctly.