When the user attempts to attach a file to a HR Case, the modal shows that is attempting to attach. However, then, the modal just returns to being blank as if no attempt was made to attach a file. Also, no attachment is shown in the header of the HR Case.
The "Read" operation ACL for sys_attachment for HR Core has been customized and is now broken.
As mentioned above, the read ACL for sys_attachment in the HR Core scope was modified by a non-ServiceNow user.
Reverting this ACL back to the Out of Box (OOB) version resolves the issue. For convenience, here is a link to the ACL: