USB disks attached to Airport Extremes are not supported for Time Machine (TM). Never were - probably never will be as Apple really wants you to but a time capsule. They may "appear" to work - but as you've found out they are dead flakey.
HOWEVER - I have the same (2TB Disk attached to an AX) and I use Carbon Copy Cloner* as my backup of choice. Not as nifty as TM but does work well for me.
* Other backup programs available.
Never were - probably never will be as Apple really wants you to but a time capsule.
That's not quite correct. We had a couple firmware versions out a while back to support this. But engineering dropped the support, so AppleCare as a rule can't support this set-up either. You can TRY to push your luck here, but eventually just about everyone will run into failure with this at one point or another.
Ironically, last night (after a thunderstorm) my wife's Time Machine (on Snow Leopard) refused to back up, citing "Error 22". Up til now, I've had no problems with Snow Leopard / Airport Extreme / Seagate disk; only with Lion. As it turned out, the USB connection between the Airport Extreme and the Seagate drive seems to have been disturbed during the lightning storm (we DO have surge supressors, everywhere). In any event, disconnecting the drive from the Airport and re-connecting it (politely, with Airport Utility) has fixed the problem and the backups are safe.
I think Hackintosh is referring to "we" as the community on this forum.
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We had a couple firmware versions out a while back to support this.
No. There was an update early in Leopard that made it possible, and many blogs, etc., claimed that meant Apple supported it, but it has never been supported by Apple.
A little history: Time Machine was introduced with Leopard. Some of the pre-release info did say it would work, and it was tested (with mixed results) during the beta period, but when Leopard was publicly released, it did not work and was not supported. Some folks discovered that they could use that configuration anyway by employing a Terminal command that allowed "unsupported" destinations. A few months later (about 10.5.4, I think), that command was no longer necessary -- apparently to allow backing-up to USB disks connected to Time Capsules and/or 3rd-party NAS drives that meet Apple's requirements. That's what some incorrectly assumed was Apple support.
Bottom line: It's usuallly unreliable. If you value your data, use a more reliable destination. See Time Machine - Frequently Asked Question #2.