I experienced this error "Word has encountered a problem trying to open the file" when trying to open a Word template (dotx) from SharePoint 2013. I should explain that I was using Word 2010 and could open the template from its location in the file system, but when I uploaded it as a content type into SharePoint 2013 I received this error when trying to use the template.
Some blogs suggested the following
a) install Office 2010 32bit instead of Office 2010 64bit
b) upgrade to SP2
c) remove the SharePoint Foundation Services as a component of Office, and then do a repair.
I tried all of that and yet it still would not open the template. Then I found a blog that advised switching off Protected mode - go to File, Options, and Trust Centre Settings and disable all the Protected options which are enabled by default. That did the trick.
My advice - try switching Protected mode off first before trying anything else. Now I come to think of it I had seen this before and had forgotten about it. Hence the blog.