I was having the same problem with an HD movie rental (at first the screen was blank, then it would start but stop and never go anywhere; I could scroll forward and see still screen shots but nothing would play). I experimented with downloading an SD movie--and, actually, the SD movie played just fine.
Later, the HD movie, in contrast, played for about 15 mins when I selected the MiniPlayer, but then it stopped and the problem returned and I could not get it to play again. I wasn't able to download the SD version of this movie because all the option icons (the buy, SD rental, & HD rental) read "downloading" simultaneously.
I had been noticing that my WiFi signal had been slower than usual in the last few days (there were a few rooms in my house where I had no signal at all the day before). So I live-chatted with my internet provider (TWC)--at like 11 p.m. because I was that annoyed that I was 15 mins into Gone Girl and couldn't finish it--and they tested and reset my internet signal & speed. And, lo & behold, that actually solved the problem! Within minutes I was able to play the rest of movie and I haven't had the problem since!
Now, if I could only figure out how to get iTunes to save my CC information as opposed to having to buy gift cards!