One thing is information and quite another is information processing. The former is much larger, indeed, than the current cognitive capabilities of society - in any time, no matter what. The latter, on the contrary, is by definition, much more symmetric to the cognitive capabilities.
In other words, information can be taken as an environmental asset that form the outset is much larger than what society can use of it.
After all, information is the concept introduced in the 20th Century to denote exactly the same that the 19th century used to refer to with the concept of "energy". With one proviso: information explains the same as energy does and still much more and much better.
Without getting into discussion about the difference between "data" and "information", the answer is yes.
Information about the weather has always exceeded human capacity to process it. Humans have produced devices intended to predict the weather since they could pile rocks on top of each other. The amount of information is growing every day. It is growing faster than IT systems can process.
This is true if we limit ourselves to external mechanisms to measure weather. If we consider that human beings are constantly experiencing weather via their bodies, which are processing the information in certain ways, there has been an unquantifiably huge amount of information about the weather than has never been recorded, and therefore escapes the most powerful computers and algorithms. And no one should doubt that this will continue far into the future, perhaps forever.
Similarly, many forms of astronomical information exists in an amount that existing resources cannot process. For some types, there are estimates that it would take centuries to process the data from certain telescopes or satellites. Scientists have turned to crowd-sourcing for years in attempt to deal with this information.
One example, probably the oldest, is the SETI program. To deal with the information volume, the general public has been requested to assist by downloading a program that will cause their computer to analyze data from the SETI program while the computer is otherwise not in use.
But to get back to what I said I was going to avoid, in many cases what is being referred to here as "information" would be considered "data". "Data" is what is collected by measurement. "Information" is created when the data are analyzed.
Under that standard, the answer is no, it is impossible, because information cannot exist until data is processed and categorized.
The amount of information has already exceeded our capacity. A mechanism that we use to manage the exceeding capacity and complexity is through abstraction. The problem then is: what is the appropriate abstraction to model the underlying deluge of information?
The answer to your question is yes. It has been so ever since humans began living in cities - and even much before.
Our brains can recall only 7 independent dimensions at a time. The dimensionality of a modern society is beyond calculation. So what we modern humans have to deal with on a daily basis is far beyond our mental capacity.
One result of this is for many people to seek and accept simple answers to complex questions.
Another result is for many people to follow their emotions/impulses rather than thinking things through.
What makes for a wise person is to pause (use her/his prefrontal cortex) to put a given question in context, to seek more information, to weigh priorities, to consider the impact of alternate decisions, etc. Then, after this pause, to give an answer - with all the caveats required.
This last approach requires training one's brain so that one avoids following our primitive human impulses that often lead to serious consequences.
Similar to robots we need all world computer brain for information processing and decision making. Maybe in the future it will be physical and intellectual communism: robots for mechanical work, the brain for management.