There's a setting on Facebook that does not allow other people to add you as a friend, you must add them.
For example, if I set my profile in this manner, Suzie Q could not add me as a friend, but I could add her and then she could confirm. I don't really get the point of the setting for anyone but celebrities or something, but some people are extremely cautious on Facebook.
If it's happening for all friends you try to add, then not sure what the problem is :/
I've heard there's more to life than an FEI tent and hotel rooms, so I'm trying it.
Yep, the person has it set up that way. Assuming it is someone you know well and they would probably want to friend you, either ask them in person or send them a FB message or email asking them to add you.