I am having the version of this problem where the EA app installs almost all of the The Sims 3 content but won't install a few of them no matter what I do.
I have tried all the recommended fixes detailed here.
My affected games are as follows just for anybody reading this for whom the information helps:
The Sims 3 Ambitions
The Sims 3 Pets
The Sims 3 World Adventures
I don't own all the packs though, so if I'm not listing one somebody else has a problem with, don't assume that I downloaded it OK.
I also contacted EA support and got what seemed to be a copy-paste e-mail with the same generic advice everybody has already followed. My guess is that the EA support folks are inundated with requests for the same issue and are thus basically sidelining this until they find something they hope will work across the board. I'm not mad at the people handling the tickets, I'm mad at their bosses, but that's whatever.
Just adding my name to the group of people that keep experiencing this issue like me.