Getting this error while editing a word document with Office web app 2013, but am able to edit and save Excel sheets. My documents are on SharePoint 2013 environment.
I currently can't add comments but here's some things to try.
Create a new word document in SP2013, open it and see if you can edit it.
If you can edit excel sheets, that to me suggest there might be something specifically stopping you editing that file like a check out.
We experienced a similar issue with a library migrated from our 2010 environment, I was able to edit the document with using office 2013, save the document back. Afterwards Office web word was able to edit the document....
I was facing this problem because the below property was set to TRUE on SharePoint 2013 Working Web Application.
$My_SITE.WebApplication.DisableCoauthoring = $true;
After setting this property to FALSE, the Word web app started working for Word Documents
$My_SITE.WebApplication.DisableCoauthoring = $false;