I have a webpage with a tag on it with the following markup:
I had the same issue with IE 11 and the problem was the content type was application/octect-stream as stated by csmith. I was serving my videos from Azure storage and apparently that is the default content type.
You can change the type with Azure storage or using Azure api as shown in the post
Set Content-type of media files stored on Blob
Here is a guide for people on Amazon cloud (I did not test it)
Changing content type for Amazon