So taking this down to a very basic level, the problem is this: your choo choo can't get to the station because there's a bridge out on the way. As long as your choo choo is leaving from the same place, and heading to the same destination, it's going to follow the same route every time.
There's only two ways around that... change where the train leaves from or change where it's going.
You can change where the train leaves from by changing your IP address. This can be as simple as shutting down your modem & router (and unplugging the cable from the modem) for anywhere from 5 minutes, to 8 hours, to 24 hours. That will work
for some people. If you have a cable company like mine, that assigns a damn-near eternal lease between your router or modem or computer and the IP address, it won't work.
In that case, you have to change the MAC address the company is assigning an IP address for. If you use a router, you need to change the MAC address of the router. This is what I had to do to get mine working again. If you don't already know how to change your MAC address, you probably shouldn't be trying this... and since I'm not willing to assume the risk of you screwing up your network by following my instructions, I'll leave further steps to you.
The other thing you can do is change where the train is going. ANet/NCSoft uses different servers for different districts. For me, it was the American districts I couldn't access - because the route from my home to the servers hosting those districts had a proverbial bridge out. For about two weeks, I could only access via the European servers... no following the bad route into Texas, no problem connecting.
In my case, I had the ability to log into my account from a different location altogether, and was able to switch my two accounts and my husband's account to a European district. From then on, I was able to connect (at least until the route to the login servers tanked... that's when I had to change my IP address by changing my MAC address) from my home again.
I'm able to connect to the American districts again now. ANet made a few internal changes that altered how the connection was traveling... unfortunately, in so doing, an entirely new batch of players became affected while many of us who had this problem previously were now connecting without issue.
EVERYONE affected by this needs to submit a support ticket -
[email protected] and you need to follow up by sending in the information they're going to ask you for. NCSoft needs this information so that they can take it to the peer that has the bridge out and beat them over the head with it. You can also nag NCSoft a bit on their Twitter page - NCsoft_support