Found on v. 4.7.9
I found that if we tried to invite multiple email addresses at the same time, if one of those emails was already a member, then none of the invites got sent.
We commented out lines 361-367 in controllers/friends.php and now the invites go through.
This bug might have been fixed in later versions, I'm not sure.
Also, a neat improvement would be to have each error'd email link to the user account, so people can send a friend request to that person. For example, "XXX@XXXXX.com is already a member! Click here to send a friend request."