One fine day I get a case (ticket) from a user. We have seen these in the past. Excel Web Access web part giving generic error. User will say their page was working and now it does not work.
The error message will be very generic. Something like:

So I embark on troubleshooting. We have DEV, UAT (QA) and PRODUCTION environments. After some aimless wondering I discovered that our UAT environment has the same problem. I found this excellent post by Wictor Wilen:
Office Web Apps 2013: Excel Web App ran into a problem – not rendering Excel files
This told me not to give up and look in the right place. ULS logs. I generated the error in UAT and then merged up the ULS logs and looked under ULS Log viewer. And found the following error:
SQL Database ‘***’ on SQL Server instance ‘***’ not found. Additional error information from SQL Server is included below.  Cannot open database “Your Content Database” requested by the login. The login failed.  Login failed for user ‘Domain\yourServiceAccount’
Note that Wictor Wilen has found out a different error than mine. But the comb up the ULS log is what I needed.
After that, I asked my DBA group to add the necessary permission to the account to the content database and we are good to go.