Actually, though... If they've been monitoring everyone's internet traffic, it's feasible they have unwittingly downloaded illegal copies of media. Meaning they've committed copyright infringement on a massive scale?
It's generally not copyright infringement to simply collect data available on the internet. For example, Google downloads tons of copyrighted information; but that's not considered copyright infringement because using it for a search engine is a "transformative" use. Google is not replacing the copyright owner by redistributing the page, but is instead using it in a completely different way to provide an index of all pages.
The same would be true of the NSA. They aren't redistributing the data they download to anyone who asks, they are instead compiling an index to be searched for intelligence gathering purposes.
IANAL, but I suspect there's a difference between uploading a video and saying "I didn't mean to" vs just copying massive quantities of data. For example offering a backup service.
A backup service means you're acting as a dumb pipe/dumb disk. You're copying bytes without knowing the contents at the request of your user. This doesn't apply when you're seeking out data to copy on your own. In some situations you're still fine, like if you're requesting files from public web servers. But I'm not aware of any protection that would blanketly apply to intercepted connections.
If you're operating a dragnet then really you're close enough to a dumb pipe. If you're doing very targeted capture then you can filter stuff out anyway.
Being a dumb service works because you're acting on behalf of someone else. Someone has to order the data to be copied, and they are the ones responsible for ensuring there is sufficient licensing.
The same would be true of the NSA. They aren't redistributing the data they download to anyone who asks, they are instead compiling an index to be searched for intelligence gathering purposes.