I considered using Access, but it was not robust or flexible enough for my particular needs. Instead, I am using User Role Editor to manage permissions.
PROBLEM:
Users receive the message below when attempting to manage many-to-many relationships:
"You do not have rights to create new items"
FOR TROUBLESHOOTING:
I provided you with access to our staging site.
All of our many-to-many relationships are effected, but you can begin with this one:
Where, teaching-assignments is the intermediate table. When attempting to edit a course-section, and then add a new teaching assignment, that's when the warning message ("You do not have rights to create new items") occurs.
BY THE WAY...
It's possible to create many-to-many relationships when directly editing the intermediate post type (in this case... "teaching assignments").
However, when I attempt to create the same many-to-many relationship from within either of the parent post types (in this case... course sections, people), I get the warning message ("You do not have rights to create new items").