Well, Snowden wasn't DoD, but yeah, several layers of encryption are usually looked at if feasible. That is the main concept by Defense-in--Depth I believe.
He wasn't an actual DoD employee, but rather a Dell sub-contractor working as DoD SharePoint administrator, which makes his access to classified documents all the more preposterous. Civilian contractors shouldn't have access to classified documents at any point in the information exchange chain. I hope that was the main take away. That and encryption.
"Do not seek to follow in the footsteps of the wise. Instead, seek what they sought." - Matsuo Basho