Airport Disk – How to access it over the Internet without iCloud or Back to the Mac

Contact your upstream internet provider (Cablevision) and let them know you wish / need to manage the network and want their device to operate in bridge mode. Often someone will post the details on how to get into your modem and re-configure it in cases where the provider doesn't want to assist you, but I've always had good luck asking (especially the techs that will have to come to your location for trouble calls.) If you call once or twice and explain you call each time this Double NAT error happens, they will quickly realize what you need to run your network without having to call them on an ongoing basis.

The Airport hardware doesn't have anything to do with this detection, so the software in the router is detecting a change in the upstream network configuration and reporting it.

Usually I've seen this with metro-wide WiFi networks where a fault will change the routing through another router that is misconfigured or not normally serving the modems of customers directly.

Unless you're changing the settings of the Airport, the cause is almost surely up stream from the device itself.

Make sure that you have a clear path to the Internet in terms of firewalls, and ensure that TCP port 5354 and UDP ports 4500 and 5353 are open and unrestricted.

Once you've done that, make sure you are logged into iCloud on your computer, and that the remote Time Capsule is on (also making sure those ports are unblocked on that end). Your disk should now show up via AFP in the Finder sidebar!

Also try these steps from the Apple Support Article about troubleshooting Back to My Mac.

• Update all your Macs to OS X Lion v10.7.3 or later (OS X Lion v10.7.4 or later is recommended).
• Choose System Preferences from the Apple () menu. Open iCloud System Preferences and check that Back to My Mac is enabled. See if any diagnostic messages are provided under the Back to My Mac section to assist you with troubleshooting and resolving any connection issue. 
• Toggle Back To My Mac off and back on by deselecting and then reselecting Back To My Mac in iCloud System Preferences. 
• Make sure the computer you are trying to connect to is not sleeping and is set to Wake on Demand. For more information about Wake on Demand, see this article.
  
• If you are using an AirPort base station, make sure your firmware is up-to-date (open AirPort Utility 6.0 or later).
• Check your network for a multiple NAT setup and reconfigure your network as needed.
• Make sure NAT-PMP or UPnP is enabled (if you are using an AirPort Extreme Gigabit Ethernet base station, click here for information about AirPort Extreme Gigabit Ethernet and NAT PMP).
• Ensure TCP port 5354 and UDP ports 4500 and 5353 are open on your firewall. To learn more about ports used by Apple products, click here.

Note: Perform these steps only if you've verified the information in the previously listed troubleshooting section is configured correctly.

1. Open Keychain Access located in /Applications/Utilities.
2. Choose the System Keychain in the Keychains list on the left.
3. Sort the Keychain list by Kind.
4. Select all keys listing the "Back to My Mac Key."
5. Delete the selected keys. You will be prompted for your machine's administrator password.
6. Quit Keychain Access.
7. Choose System Preferences from the Apple () menu. Open iCloud System Preferences.
8. Toggle Back to My Mac off and back on again by deselecting (if still selected) and selecting the checkbox.
9. Once Back to My Mac connects, quit System Preferences.