I had a problem with Instagram videos, luckily I was able to fix the problem pretty easily. Here's how:
Fixing Instagram videos if they won't play
Task 1: Make sure you are running the latest operating system on your phone. If you are on Android (and there are known issues with Android support on Instagram) then please make sure you are running 4.4 KitKat.
If you have an iPhone, then iOS7 or above is preferred (the latest version is always better - as it will contain more bug fixes).
Task 2: Check that your Instagram app is actually up to date. Even if it is, sometimes it is worth deleting and reinstalling Instagram all together.
Don't ask me why that works, but it fixed my problem with the Instagram app, so I think it might be helpful for you too...