I know this is an old thread, but I thought I would add some technical info to help clear some things up.
So what Tezgno wrote made sense to me except for one thing—I figured the cable modem used NAT so that it didn't matter what router (or computer) was plugged into it, and that you would always have one public IP address. I guess that's not the case. Any device plugged into my cable modem generates a new public IP address (at least with my ISP).
If you plug in a Airport Extreme (AE) into the cable modem, it gets a public IP address, and any computer attached to the AE base station gets the same IP address. If you bypass the AE base station with your computer, then that computer will get a brand new public IP address.
That means if your AE address got its IP banned, it's possible that if you get a new router, the same thing may happen again, which is what happened to me.
I got exactly the same problem a few days ago, about the same time my Mac had a hardware failure (2011 MacBook Pro 2nd graphic card/mother board failure, another known magical issue). At it was failing, it messed up the email accounts to my websites. Due to multiple failed login attempts, my own websites blocked my AE's IP address. Bypassing the router helped, until the bad email accounts tried to log back in to my websites, and then that new IP address got banned as well.
I had called my ISP (WoW), who reset the cable modem, which reset my IP address, and temporarily worked. When the Mail program ran again, my new IP got banned.
I called my website host, who unblocked my IP addresses (now many addresses, due to multiple AE base stations that I tried) from my websites, and then I fixed the email accounts on my computer. The problem seems to be solved.
I've used many routers in the past (D-Link, Netgear, Linksys, TP-Link), and always went back to the Airports because they've never had an issue (including not this one, apparently). I've owned every version, including the Express. I had one Express fail (not plugged into surge suppressor in a storm), and one pizza-box fail when I set the latest tall AE on top of it for a few weeks. Even though the pizza box was not plugged in, it refused to turn on afterwards. I think the new AE fried it being in close proximity for so long. Something to thing about as these signals go through us continuously.
So at least in my case, Tezgno had the correct diagnosis.