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NotStan said:

Once you recover an account to the xbox the same account on a difference xbox goes dead, if you earn any achievements on the other account that is dead they won't count. When you try to sign onto xbox live on the old account you won't be able to due to some error screen coming up saying that you need to do recovery again, so you can only connect account to 1 xbox at any given time, and if you want to connect on some other xbox  you'll need a USB stick or recover the account. If you recover the account and the old xbox goes online and syncs with the XBL servers you can't play the XBLA games on that old xbox anymore.

So to conclude, I think if the old xbox stays offline there shouldn't be any problems of playing XBLA games on it offline.

that sucks

this post seems to help the most