We're using video.js which uses vtt.js and from a Samsung Smart TV, the subtitles are large and overlap the video

As mentioned here, it seems that it could be vtt is not calculating the sizing correctly in full screen mode.
It's difficult to debug, but to replicate If you go to http://videojs.com/advanced/ and select the 3rd video "Elephant's Dream" and enable captions, this has the captions same as above - large and on top of the video.