I'm serving .mp4 videos and .mp3 audios using private download (field's upload destination: Private files), and they play OK on native html5 player, but seeking doesn't work, as in I can't jump to the middle of a video or audio.
This happens on Chrome and doesn't happen on Firefox. Also no problem playing and jumping on public download or when the file is saved on local drive and played from there.
After comparing the response headers in several different scenarios I have concluded that the only consistent difference between working and non-working is that Accept-Ranges and Last-Modified headers are missing when seeking is not working. These can be added at hook_file_download but then the media won't play at all.
This a pain to test with Chrome Developer Tools > Networking as Chrome pulls old headers from its memory, so you never know which input is producing which output.
Install https://www.drupal.org/project/resumable_download. Despite its name and the fact that it's at beta1 (it gives some notices) it does the job. It takes completely care of feeding the media to the browser.
Edit: If the module doesn't work, check out this patch: https://www.drupal.org/node/2362393