Xbox 360 - Plants vs Zombies - unable to download update and won't play

The "You already have as many items as you are allowed to have try again later" message, is that an ingame message or an Xbox message? If you check the Plants vs Zombies Garden Warfare hard drive space, do you have any corrupt/incomplete downloads that you can delete?

After the above, I would suggest either try clearing your cache or delete the game and try redownloading it.