I mean, that's hard to believe. I work for a large storage vendor. If this was true, the feds would be our largest customer by far. They aren't. I don't think SE Asia makes enough disk to store everything indefinitely, and given exponential network traffic growth, even storing the most recent few years is a problem.
https://en.wikipedia.org/wiki/Utah_Data_Center
The explosion of data you mention is a real problem; though thinking about it, one might manage with a mix of light filtering, leaning on organizations that already store the relevant data, etc.
In your comment, for example, you seem to suppose that those agencies do not buy storage on secondary markets. Can you confirm that? Because if they do, then your sales statistics are almost meaningless in this regard.
Also, it seems like you assume that everything the agencies collect is stored "as it was transferred" (i.e. if the same YouTube video was viewed millions of times, they would store it in millions of copies). I think it is safe to assume that these agencies are a little bit smarter than that and if they store everything, they have some kind of compression in place to minimize redundant storage of the same chunks of data.
In any way, I think that until someone like Snowden speaks up again and brings evidence from inside of these agencies, we are only left with speculation.
We don't know whether the NSA does "Full Take" today. That's speculation. I would be astonished if they didn't, and it is reasonable to behave as if they do.
It's possible, it was possible 4 years ago, and storage is not getting _more_ expensive over time.