Steps to reproduce...
1) Add to functions.php:
add_image_size( 'half-size', 200, 200, false );
2) Add to template file (inside loop):
<a href="<?php the_permalink(); ?>" title="<?php the_title_attribute(); ?>" > <?php the_post_thumbnail ('half-size', array('class' => 'profile_image')); ?> </a>
3) Go to media library and add image to post gallery. Then crop image, click "thumbnail" radio button and click save underneath the image.
4) Click set as featured image then "save all changes" at the bottom of the gallery dialog window. Then update the post (may not be necessary).
Do not insert into gallery. Click over to the actual post and see if it is showing the part of the image you cropped or the whole image.
What am I doing wrong? Or does cropping inexplicably not work with Featured Image?
Apparently, there is no ticket out for this issue.