Yes. But that doesn't mean good historical work can't be done.

Perhaps premodern historians and medievalists are in a position to read every single relevant primary source that might be even tangentially related to their topic -- even if the secondary literature is probably still daunting. But many modern historians have found themselves in the position of dealing with an overwhelming volume of information for at least a century. In fact, I'd suggest that acquiring the skills for dealing with "too much history" is one of the most critical skills historians learn in their training.

I'll use myself as an example. My dissertation deals with the twentieth century history of a single Micronesian island. This island has no written language. Surely I should be able to get through all of the published and unpublished material related to this one island, right?

But here's the problem. The island was occupied by Spain, Germany, Japan, and the United States during my timeline. That means that are relevant archives in Madrid, Berlin, Canberra, Tokyo, Washington, San Francisco, Honolulu, and Micronesia. The largest of those archives has 2,000 reels of microfilm and I believe something like 3 million pages. There are newspapers I'd like to look at, the papers of anthropologists who've worked in the field, published documents, diaries, travel narratives, scientific reports... the list goes on. And of course I need to go and live on the island again for a while to gather oral history narratives. The extent of knowledge there is practically bottomless. So even with a project that may seem easily delimited, sources quickly become overwhelming once you start to look for them. The trick is figuring out how to manage that source base anyway.

Can I add a follow-up question to OP's? It's about pedagogy, so if it doesn't belong here, please tell me and I will remove it.

Recorded history only gets longer. Who determines what gets included in mandatory K-12 history/social studies curricula? What sorts of things have had to be chucked to make room for other material? What topics/time periods/events are on the chopping block, soon to be excised? Is anybody here a high school teacher? Or do we have anyone here who advises at the state or municipal level regarding curricular selection?

I'd like to offer a slightly different perspective here: just because records are kept doesn't mean that they're going to get preserved.

It takes a large commitment of time and resources to preserve something for posterity, especially when whatever that thing is might not seem immediately useful. Why, for example, would someone want to preserve an old computer, even if their browser cookies might tell us about popular trends on the Internet? Why would someone keep their receipts or their bank statements beyond the time they need them for tax purposes, even if they might tell us something about consumer culture? Why would someone keep digital photographs they don't want, even if they might reveal things about media culture, and especially considering how easy it is to erase them and take new ones? And even if you personally deem them worthy of preservation, will someone think the same after you're dead?

Maybe you'll decide to donate things to a local archive or museum. It seems, however, that now is not a very good time for either archives or museums. Economic problems often place them first on the chopping block for funding cuts, and part of that involves how much space and money they can dedicate to preserving material. Some archivists have to make very tough decisions about what to keep and what to - literally - throw out because they can't afford to keep it all. And we're not just talking about stuff that people donate, but official state records. Private companies sometimes do the same thing with their archives, especially if they're going under. If Google goes out of business, who's going to pay money to keep their servers running so they can preserve all the data they've collected? Sure, someone might be interested in buying it, but all of it? Or only the stuff they find useful? And that's not even mentioning situations that could otherwise physically destroy records. What if there's a fire/flood/earthquake/civil war? How many people will think the first thing we need to save is the archives?

I don't mean any of this to seem facetious, because it's a very serious situation that a lot of people who work in the industry have to face every day: what, exactly, is worth preserving? And they don't always get to decide that. We like to think with the Internet that nothing ever really gets destroyed, but Web 2.0 has been going on for - what, maybe ten years? Fifteen? What is it going to look like in 30 years? 100 years? 1000 years?

98 percent of the records produced today are considered junk and are usually forgotten or trashed because people both don't know what's going to be important to future historians and because they don't have the resources to preserve everything. Even modern historians have to run up against a lack of sources in some respects, and it only gets worse the more time passes.

So again, to reiterate: just because we're producing more doesn't mean we're going to keep it.