"the NSA would need a massive system to handle just the incoming data let alone one to sort through it.
It's more likely that the NSA just has Taps into the lines and can sort through the streams as they pass by."
You're really talking about the same thing. There isn't one "big pipe" that AT&T just hands off to the NSA and says "here ya go!" There are probably several colletion points at the key AT&T core backbone nodes (Chicago, DC, Dallas, etc.) Circuits from these points are most likely passive taps.
The NSA is more than capable of handling AND sorting that amount of data (AT&T public backbone traffic) and then some [understatement.] Even if they're not getting direct feeds from other backbone providers, you can bet they already have taps on other core fiber routes around and the leaving the country (submarine trans-Altantic/Pacific, etc.)