Why aren't some of my songs transferring to my new iPhone?

I purchased a new iPhone a couple days ago. During the setup process, I restored from a backup of my old iPhone, from about two weeks ago (the old iPhone was stolen, so the backup was a bit old). When I restored, I got a warning that 21 items could not be synced, and to check iTunes for more details. iTunes did not give me any warnings or information at all.

I noticed that the 21 songs are all songs I had previously purchased from the iTunes Music Store. They are not on the iPhone; in the iPhone music list view (in iTunes), they all have a sync icon next to them:

I can't seem to get them transferred to my iPhone. I tried removing all the music from the iPhone and resyncing, but they still were not transferred. I then reauthorized all my computers, and re-authorized my computer, but the songs would still not sync.

What's wrong? Is there a way to sync these previous purchases to my iPhone? I should note that this is not happening with all of my songs; some purchased songs were transferred successfully.

After trying a series of troubleshooting moves, including restoring my iPhone from an iCloud backup, the only thing that worked was restoring my iPhone from an iTunes backup, as described in this article.

I was having a similar problem and had to go into my iphone's settings, under music, and switch on "Show All Music". The songs showed up in my playlists and then I just had to click the cloud to download them.

That totally works!!! Go into iTunes on your phone then into purchased and hit 'not on this iPhone' at the top of screen. Then start downloading all of your missing songs!