Exactly same problem

I had the same issue and just resolved it by switching the enclosure. LaCie tech support sent a new power cord, then stated I'll I could do was send in the entire unit for repair, with me having to pay for data recovery services; they were unwilling to repair/replace the enclosure without reformatting the drives. Instead, I purchased a new enclosure and swapped my old drives into the new enclosure. The drives showed up, volumes intact, with all data accessible. So the problem was contained in the enclosure but the drives were not impacted. Side-note: I was told by tech support that the new version of the same model would be incompatible with the old drives - not sure if true or false, since my fix used the same version / same model.

Check your network for conflicts, check power supply, fan and nic card, if your sure it's not the the drives inside the drive case, I have had 4 of these fail on me @work with different issues power supply nic card and both drives 4 units did the same thing and they were under warranty so when I got it back, I sold them and bought The Seagate BlackArmor NAS 420. no problems with them as of yet. Lacie get hot and components get hot and fail because there is not enough air flow I think.

I had the same symptoms, only my light remained blue. Drive did not power down and was not visible on my router DHCP table.

I first recovered my data using following the instructions in the second post on this thread:
http://www.linuxforums.org/forum/peripherals-hardware/126831-reseting-lacie-2big-network.html
It involves mounting the Raid partitions onto a Linux system.

I then found there is a reset procedure and called Lacie for them to confirm that. This is the procedure they sent me.
I executed that and now my router sees the drive.
Still not sure if the thing works though. I did this with only one drive in its bay just in case.

Once that is done you need to recreate the original shares on the admin page for the drive. You need to do this to access the files.

It took me a lot of investigating and work to finally get to my data.
Lacie phone tech assistance wanted to send me a new power supply (obviously this is there most common problem).
I had to tell them I new about a reset procedure to get them to send it to me.
They were basically of no help at all. There suggested solutions consisted of lossing my data by sending the drive in for repair, or paying loads of money for a professional recovery.

This is the reset procedure:

The procedure below will reset the 2Big Network to factory settings:
IMPORTANT NOTE: Following the procedure below can ERASE ALL DATA on the drive. Prior to following these directions, verify important data is stored securely on a different storage device.
Switch the drive OFF. Once it has shut down, ensure the power switch remains in the OFF position.
If the drive is in the SAFE mode, move the RAID switch to the BIG mode.
PRESS AND HOLD the button on the front. MOVE the power switch to the ON position. The light on the front of the drive will start blinking.
Once the light stops blinking, RELEASE the front button.
When released, it will begin BLINKING AGAIN. When it stops, PRESS AND HOLD the front button. The light will begin to blink again.
When the light stops blinking, RELEASE the button. The drive is now resetting itself. Switch the RAID selector back to SAFE if desired.
To restore data if it was lost during the process, recreate the shares and users on the unit prior to the reset. Any firmware updates applied will need to be applied again.