just tested to see
turned on Steam downloading a game, then turned on my 360. from the time I pushed the power button to the time I was on the dashboard was about 1:15. so that's definitely shorter than my 2 minute estimate.
that 1:15 got me to the dashboard with an error code about not being able to sign in. so presumably if I played a console that requires you to always be online, I'd still be out of luck. for online games that doesn't matter, I wouldn't be able to play online anyway with the internet in use like that. but for a single player game, that would kind of suck!
on the bright side, if you can't sign in to XBL the games from this gen consider you offline and don't try to talk to a server or whatever most of the time. so you can shave about 30 seconds off the "get to the main menu" time for a game like 3S for example.
if you get the worst possible outcome it's that your 360 will connect to XBL despite the lag. so you wait that ~1:15 and you've got another long wait as whatever game you choose goes through the motions as well.
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http://www.youtube.com/watch?v=npHDxSvwCE0