Hi Crozy,
Essentially when your Airport products hang with a solid amber light, they are stalled during the 'boot up' phase. It is a concern if multiple units are developing this behaviour.
If a factory reset doesn't shake out the issue, usually hardware or firmware is suspected. I would have suggested checking to see if plugging the unit directly into an outlet works, to see if it is not getting enough power. But the Genius would have troubleshot for that by plugging directly into an outlet.
I don't think you are doing something wrong, but multiple devices developing the same issue suggests something environmental - something in the relationship between the unit and other devices.
How have these devices been set up initially? With a Mac or a PC or iOS 5 device? Also, what version of Airport Utility is used? If doing a factory reset allows you to at least return it to a default state, make sure your Airport Utility is fully up to date, configure it with a direct ethernet connection to the computer rather than wirelessly, and ensure the latest firmware update is applied to it. If you are using a PC, turn off your antivirus software and firewall temporarily during configuration. This will help ensure no errors are introduced to the firmware on the device.
Next, also try resetting the modem or router it is connected to. If using a cable modem, press the reset button on the back of the device. For DSL modems it is a bit different, if you do a full reset you may need to contact your internet service provider for help with re-entering your internet account and password credentials with them in the modem itself. Also check with them or the modem manufacturer if there is a way to ensure the firmware on the modem is updated; the modem firmware may be causing errors in the Extreme when it starts up and reads connected devices.
Check your ethernet cables too. If you have an older cable connecting the Extreme to the modem, try a fresh one, and also make sure the cable you connect to the computer to configure the Extreme is also known-good.
Lastly, ensure no transmitting devices are set next to the Extreme - cordless phone base stations, etc. and nothing set on top of it. Radio-frequency waves in direct proximity could potentially set up RF induction, in which the radio signal passing through circuits can cause an electric current to occur. Objects on top of the unit could cause overheating.
I wish I could be more specific but those are the factors I think could be contributing to the issue... I hope at least one of them helps. A very unusual situation, for sure.