I have the serial number and restore code and have checked them.
I have internet access on the phone and have tried from wifi (to adsl2), 3G and LTE networks. Other internet based applications are working.
I have been persistently getting either "A network error has occurred" or "Server is down" messages for the last 14 hours.
I haven't seen any relevant threads on the topic in here or anything in the Authenticator FAQ or knowledge base.
EDIT: just in case it was related to the phone, I also tried on my iPad, same error.