I'm having a similar problem but its only on certain wifi networks. If I'm in my apartment on my wifi network, my iPhone 5 and iPad mini, both on 6.0.1, will not backup automatically or manually. However, if I'm at my parent's house on their wifi network, both devices have no problem backing up at all.
Do you also have problems sending out a picture iMessage while connected to your network? I always have to send pictures as a text message while I'm connected to my network or else they will not go through.
Also, do you have an airport express, airport extreme, or time machine?
The reason I ask is because the only thing I can think of that is different between my wifi network and my parents is I have an airport express while they have a Belkin router so I'm wondering if that is the problem.