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I only experienced a short period where the TV app was unable to broadcast, but was able to watch Obama's speech in it's entirety. Didn't have any problems with email. Received three of them during the broadcast.
I don't know where you live, but I had this same problem yesterday. I live in So Cal and I couldn't access my email all day. It wasn't until about 7 at night that I started getting mail again. I did notice today that the TV was down today, but I figured that was because of everyone trying to watch the inaguration speech.
My email is out frequently, the funny thing is when it does finally start working again, it'll indicate 50 new messages, though I may only have a few. Wierd glitch that is.