The phone does not "dial" the emergency phone number. It tells the tower that it needs to initiate an emergency call. At that point, the infrastructure bypasses all authentication/authorization/billing checks and connects to what the infrastructure knows as the local emergency service. It's a standard voice connection, made over the same channels as a normal call, with one exception: if there is no available channel, it drops a normal call to make room.
In this case, some of the potential situations could be occurring: 1) There is still backbone (backhaul) availability, but perhaps the connection to the subscriber database has been severed. No normal calls can be authenticated, but emergency calls don't get authenticated and are thus not impacted. 2) Everything is working normally but at drastically reduced capacity. With the normal load of emergency calls, there is simply not enough capacity for normal calls. If this is the case, it is possible that not all emergency calls are being completed - the emergency operations center would not have the ability to know if this was the case.