"Glad" to hear I'm not the only one. So this seems to be a problem "introduced by Apple" and not really in my network settings.
Since not everybody has this problem I assume that the Apple ID Server has a very distinct problem. Could be reacting on log in attempts coming from a MacBook Pro (Retina Display Mid 2012) via Wifi over an Airport Extreme for example...
I manage to log in after several tries (1 successful login out of 5 tries) but after being logged in none of the iCloud Music services work properly. See my case description here: iTunes account login doesn't work - I've tried it all...
I wonder if it is the same in your case... as soon as I connect my Macbook to the internet via my iPhones personal hotspot (instead of Wifi via Airport Extreme) -> the log in immediately works and the iCloud Music service is fully functional. So trying via another internet access point might help. Surfing over the iPhone hotspot at home is no long-term solution of course...