Getting this error: You cannot modify this Taxonomy

I have created custom posts and a role with minimal permissions – just enough to add and edit those custom posts.

Logging in with that role, I get an error message on the categories, post tags and custom taxonomies: You cannot modify this Taxonomy

Digging through the files I find that this error message is triggered by the following:

<?php if ( !current_user_can($tax->cap->assign_terms) ) : ?>
	<p><em><?php _e('You cannot modify this Taxonomy.'); ?></em></p>

How do I ensure that the current user has the capability to assign terms?

I’ve created the custom post type using code, and I’ve used code to create the role. I’ve also used the Members plugin to create the role. I’ve not yet solved this problem from either angle.

I’ve added and subtracted a host of capabilities to the role, and none of them seems to make the slightest difference for this specific problem.