I am posting to say "me too", because I get the feeling that this is happening to thousands, perhaps tens of thousands of people and not only does Apple not acknowledge it (via Support or otherwise publicly), but they do not have some kind of fix available after several days of outage. Apparently they have adopted the policy that unless people complain about it then it must be alright. So I'm complaining, and I hope that everyone else who has this problem does too.
The firmware update in p1 that folks refer to here (iPad2,1_6.1_10B141_Restore.ipsw), is exactly the same update that iTunes downloaded for my iPad on January 31, 2013. It is the only firmware update in my iTunes update directory (~/Library/iTunes/iPad Software Updates/) and, in addition to a manual check of byte-sizes of each image, cmp(1) confirms that the two binaries are 100% the same.
Therefore, it is no surprise that the update performed by choosing the manually downloaded image when chosen (via SHIFT on Windows, Option on Mac) fares no better than the default iTunes procedure that uses the same image in its own update directory.
I have verified with a packet sniffer that my Mac is successfully contacting the Apple servers (hosted by Akamai), and that there was no problem with my network, power or any other factor each and every time I have attempted to complete the update to IOS 6.1 on my ipad2. The number of times is well over 30 times during the past three days.
My computer had run out of space earlier in the week so there is no backup image in my Mobile Backups directory, and the updater for 6.1 neither advised nor provided its own backup of the iPad before proceeding. Now I fully expect to have lost all of my recent data, but I would accept that in order to be able to use my product.
This is a genuine bug, it should be quite easy for Apple to reproduce and then acknowledge while they work on a solution. I don't know how this made it past QA but it is now time for Apple to take some action, starting with communication to its ipad2 customers.