WordPress 3.9 not using custom template

Since I don’t have your theme I can’t find what causing this problem, but this should definitely solve your problem for now

Create content-{post_type}.php file and add your custom-post.

in the index.php add the below code above <?php get_template_part( 'templates/content', 'format'); ?>

<?php if(!in_array(get_post_type(), array('post', 'page', 'attachment', 'nav_menu_item', 'revision'))) ?>
<?php get_template_part( 'content', get_post_type()); ?>

or if you have only custom post type create content-{custom-post-type-slug}.php template file and this to index.php inside loop

<?php 
if(get_post_type() == '{custom-post-type-slug}') :
get_template_part( 'content', {custom-post-type-slug}; 
else :
get_template_part( 'content', 'format'); 
endif; ?>

NOTE: This is only for temporary purpose, until you find a real good solution or why the default template-hierarchy is not working.

EDIT: Install the debug-bar plugin I think you will find what causing the problem with this plugin.

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