Snowden, a US citizen, chose to leak the documents to Guardian and not New York Times, meant to be published while he is in Hong Kong, which is under Chinese rule in the matters of foreign policy.
I'm genuinely trying to understand as to why he didn't trust a single person from his country? With the claims of the access he made, it should have been far easier to find one, without going to a non-US newspaper* and boarding a plane to a nation that is not on exactly the friendliest terms to US.
(* Nothing against Guardian, it's just that the visible pieces of the puzzle do not seem to fit together.)
It is very hard to believe that he had sweeping access levels. For one, NSA has its own chip manufacturing facilities, teams of mathematicians and security experts. Even if we don't know how NSA works, we can deduce from its contributions to open source: selinux, Apache Accumulo. Surely, they are expected to have tighter systems.
Besides, he was an employee of a contractor. Even less reason to provide elevated access to him. So, if he has material in his hands, than his security clearance would allow, then it is way more surprising.
Snowden's statements ought to be taken with a grain of salt.
Edited: grammar.
Update: thanks to zecho and brown9-2 for their comments; Snowden leaked it to an American.