When storage and network bandwidth increase sufficiently, when we start measuring storage costs in terabytes rather than gigabytes, the sheer size of the data set is no longer going to be a preventative measure against ex filtrating it.
What are the chances that either corporations or 3 letter agencies are going to voluntarily delete their data on you before we reach that point?
So there will be a need to upgrade wholesale bandwidth just because the internet keeps growing, people are already talking about next gen game consoles to be cloud based (ie the rendering to be made remotely) which will add more traffic. But not 5 orders of magnitude. The amount of data stored by the gmails and facebooks of this world are mind blowing.
Another point is defeating monitoring. I am sure the NSA (or Google/Facebook) could not notice 1GB of upload, but I like to think that uploading 1PB of data would make all sorts of red lights flash in they network security control room.
Most bandwidth is used sending many copies of the same content. Attackers aren't going to be interested in downloading the popular video 100 million times, they're just going to grab the logs which are nowhere near that size, and although large it's not implausible that even the best security teams wouldn't notice until it's too late.
There is no hard rule that the leaks need to come from the same central database either. That is unlikely considering the fact that large scale services are already, and necessarily, distributed. Imagine thousands of attacker hosts receiving from thousands of compromised hosts.