Why can't I download certain songs/apps onto my iPhone?

I just got a new iPhone 4s, and I was super excited to put all my music on it so I could put more apps on my iPad. The trouble is, when I tried to download my songs onto my iPhone, a lot of them came up with an error saying "... could not be downloaded at this time." and then either Done or Retry - there was no explanation of why. So I pressed retry a couple of times, to no avail, and clicked done. Then it started happening on a whole host of other songs, and now I have a music library with a very pitiful selection of music on it.

I tried ringing Apple Support, but the best they could do was tell me to sign out and back in again, which didn't work, then turn it off and on again - still didn't work. The guy also seemed to have the impression that I had just bought these songs - they're already on my account, I just want to download them!

I have literally no idea what to do. Has anyone else had this problem, and managed to resolve it?!