Note: all of the songs in my library are ripped as mp3 from my cd's, so they are DRM free.
I also tried to buy a song from the itunes app on my iphone 5, but when the download is nearly complete it says that it cannot be downloaded at the moment. (I have retried it several times, same result)
It seems that I cannot get any music files onto my iphone somehow.
I found the solution. iCloud was the problem.
While configuring my new iPhone 5 I (of course) turned on iCloud to sync my contacts and other data.
However, this caused itunes to stop working with my new phone. So I restored everything to factory settings and when I was asked to use iCloud I didn't select it but instead chose the option that this is a new device.
Now I was able to get music on my iphone manually or by syncing again.
Then I configured iCloud again to get my contacts back on my phone and everything works like a charm now.