I've been trying different things for the past half hour to connect the Sims 3 games I got from the Humble Bundle to the Exchange, for one piece of content that probably isn't worth it. I've tried registering both the 16 digit and the 20 digit codes, and both of them were apparently invalid. None of the help articles were useful to me, all of them saying how to redeem the codes on Origin, as opposed to this situation. I don't know if this is a problem everyone who bought the Humble Bundle got, which EA may or may not be trying to fix, or if it is specific to me. If anyone had any ideas on how to solve this, that would be nice.

You're a bit vague on this, but If the issue you're having is that you have redeemed humble-bundle origin codes, downloaded/installed the game through the Origin client, and upon setting up your "sims user profile" you're being asked to "registering your game, to get full sims profile access and the free rewards for registering. BUT when you look in your Origin games menu for the "serial number" (and finding what's listed as a "product code"), and you try to register that number, the Sims site tells you the number is no good.

Then Yes, I am having that problem too. Sorry, I have no idea what to do about it. I just thought that IF you're referring to the same issue as I am getting, you'd like to know that you're not the only one getting it.
(if you're issue is different, and I'm misunderstanding, then just forget I said anything)