Examining the attachment.html template for the prosilver style, I realized that phpBB does not have markup that supports the HTML 5 <video> tag. If a multimedia file is encoded to the H.264 specification (.i.e. a .mp4 file) it can be encoded inside a <video> tag and it can be run natively by the browser instead of requiring a plug in.
Currently phpBB will expect the browser to have the Quicktime plug in installed and provides markup for Quicktime to play .mp4 files. Quicktime does this inefficiently and these files do not appear to stream, leading to an suboptimal viewing experience.
If attachment.html contained markup that recognized the media was .mp4 and the browser could support it, markup with the <video> tag could be added fairly simply.