How to create a custom template for entities? I'm having a dickens of a time overriding entity.tpl.php to create custom themes for my entities.

I am using ECK and Inline Entity Form. I've created entity section with bundle masthead. I've linked this to a node.

In a subtheme of the bootstrap theme, I then try to create a custom theme for this entity in a file called entity--section--masthead.tpl.php (note: I've also tried section.tpl.php and section--masthead.tpl.php and all sorts of permutations). Clear Cache. Nothing.

The only documentation is a few years old, and one of the most recent comments says this SHOULD work without the need for various hooks.
You don't need to add entity string to the template file name. These are theme hook suggestions for entity built with ECK (assuming your entity and bundle names):


section.tpl.php 
section--masthead.tpl.php
section--masthead--full.tpl.php
section--[ID].tpl.php


As you can see there is a naming conflict because section is a drupal builtin template. It would be safer to use different name for your entity.