Got it, the Sims 3 product key doesn't work for me...
Sims wasn't working quite right for me either. I got it to install and activate on origin but to create a Sims account the key I got thru Origin wasn't working. Went back to my Humble Bundle page and it acted like I had never activated Sims and gave me new keys that worked for signing up.
Everything works fine until I want to connect to my accound when the game has started. It asks me for the product key to register a new account but the product key isn't working...
That was my problem exactly. Try the one that you get from the Humble Bundle (that's what worked for me) or the key that it says you have when you go to further information about your game when you right click on it in your library.
Tried both... Still not working.