That's a great question. Short answer: digital sources are readily handled by digital tools. While the amount of digital information is growing rapidly, the power of the tools we have for examining this information is already powerful, and it's fair to imagine that they will grow much more powerful in the future.
Longer answer:
let's start by broadening the frame, one can ask-- "when does history start having more sources than a historian can become familiar with in a professional lifetime?"
Some ancient history is relatively easy when it comes to written sources. A classicist can read all the important Greek and Roman texts, though s/he won't be able to get through all the archaeological materials and inscriptions. But students of Sumer, Assyria and associated civilizations writing in cuneiform-- they'll not get through a fraction of the roughly 1 million tablets found so far. Indeed, a lot of them aren't published, so they'll have a hard time even making themselves aware of all of them.
By the time you get to the early modern era, volumes of written material begin to overwhelm-- any reader in, say, European diplomatic history doesn't have a prayer of reading through, say, all the material relevant to the Napoleonic Wars in national archives, not to mention memoires (seemingly anyone who did anything in these wars left a memoir). There are vast archives of banking and tax data, for example -- read only by a very few, if any. "Traditional" history offers all sorts of examples of enterprising graduate students going to the archives and discovering something that no one had bothered to read previously.
And now you get to digital. The digital legacy is immense. One of our most interesting early transects of the digital record is 9/11; this is in the millions of records. I suspect that if you set out to listen to recorded news reports from 9/11, you'd be watching tape for the next few years.
So what can a historian do with this?
Read them all? Watch every news report, read every tweet?
Nope, not humanly possible.
Fortunately, nearly all digital materials are either machine readable, or easily made to be machine readable. The academic profession of history once argued in a fashion that was some fusion of legal reasoning and narrative; I'm thinking of Leopold von Ranke, the father of our modern footnote (which I wish Reddit had!). Generations of history grad students clutched their collections of notecards, painstakingly copied out of sources, and assembled them into dissertations, mostly an artful assembly of testimonies, but less commonly with statistical power.
That happens much less in modern academic history. Becoming popular with Marc Bloch and the Annales, historians began to engage the quantitative, the statistical, the demography; the Annales weren't the first, but they moved the idea of history a field of inquiry with some parallels to epidemiology and economics into currency. A historian doesn't need to read every 9/11 blog post to extract meaning from the corpus-- there are all sorts of analytical tools for "scraping" sentiment and content from plain text. I would expect that this Reddit post (along with every other) is being scraped, and filtered for content, context and tone. A future historian need not read every post to have its [lack of] value surfaced by digital tools.
Future economic historians will have access to extraordinary data-- tax records, for example. Generations of historians have worked fruitfully with just one tax roll, the Domesday Book; the vastness of tax data for future historians is hard to imagine. Similarly kinds of secrets are hidden in banking and financial data. The French historian Emmanuel Waresquiel discovered records of payments by England and Austria to Talleyrand, for example-- a remarkable achievement by a 21st century historian working with 19th century records. It will be a much easier task for a 22nd century historian looking at 21st century tax and banking records; unlike Waresquiel s/he won't have to go digging through dusty old volumes in the Amsterdam Municipal Archives to find payments by foreign powers to suborn venal politicos.
Contemporary historians are already doing this with historical materials. Philologists were building frequency lists of words, using this to infer foreign influences and so on, long before digital methods made it really easy. Digital tools are already widely used, but are rapidly expanding their application in fields like "digital literary analysis". As just one example: given a reasonable amount of text, digital tools are very good at matching unknown authors against other text, identifying them-- so a future historian is likely to have a lot of information about the identity of anonymous posters and authors (including pretty much everyone posting to Reddit).
Sources:
CyberResearch on the Ancient Near East and Neighboring Regions: Case Studies on Archaeological Data, Objects, Texts, and Digital Archiving - a compendium of articles illustrating the use of digital tools in evaluating ancient materials. Note that this is a harder task than working with digital native materials, all sorts of information in the cuneiform tablet isn't immediately digitally accessible (some cuneiform scholars are able to identify the "handwriting" of particular scribes, for example)
The Footnote: A Curious History - a book by one of the historians I most admire, Anthony Grafton. A beautifully written and erudite introduction into "how historians practice history", the development of tools and techniques which while not digital, are often amenable to digital extension.
Library Accepts September 11 Digital Archive, Holds Symposium - the US Library of Congress has a 9/11 collection, the last time I checked, it had some 150,000 individual items
Informatics and New Philology - a 1990 article, not the tools you'd use in 2018, but to give a sense that people have been thinking about this topic for quite some time.
Mining writeprints from anonymous e-mails for forensic investigation - just one example of the ways in which digital texts can yield additional information to digital investigation.
Invisible Agents - a recent book by Dutch scholar Nadine Akkerman, who went into British archives and read long forgotten letters, demonstrating a heretofore unsuspected role for women in 17th century espionage. This work would have been far _easier_ if the material were in digital format; as it was it took an enterprising graduate student to go to archives, read centuries old letters in difficult hands; that's a lot more work than a historian working with digital materials has to do.
Hope & Company Archives - Just one example of the massive pre-digital data in archives that few if any read. These are the records of celebrated Dutch bankers of the 18th and early 19th centuries; in recent years a few enterprising historians have made big finds in these files (cf Emmanuel Waresquiel on Talleyrand's financial relationship with the British); but most of these files are unread and unknown outside of a handful of business historians. Digitization would instantly expand their accessibility dramatically.
QUANTITATIVE METHODS AND COMPUTING FOR HISTORIANS - Syllabus for History 590 at the University of Albany, assembled by Gerald Zahavi; this gives you a good idea of the tools an historian can access today. If you were to imagine a historian a century from now, these will look trivial.