I'm going to go against the grain here and say no, it's not entirely true. >:]

Assuming the number of pages in Principles of Neural Science by Eric Kandel, James H. Schwartz, and Thomas Jessell is a satisfactory representation of our cumulative understanding of brain function, it looks like after the 2nd edition in 1985, our knowledge has only been monotonically expanding. In order for the quote to be true, there would need to be an exponential explosion of knowledge in the last decade and a half.

If we're generous and combine the knowledge gained since the last two editions (over 20 years of research), the total number of added pages is just 569 relative to 1191 pages of "old knowledge." Even if we did try to account for older material that could have been edited out – whether due to being outdated or greater conciseness – it's very unlikely to shift the ratio in the other direction.

Of course, the Principles of Neural Science contains only the fundamentals, so page counts won't include our understanding of the "thinking brain" based on working theories. Also, there is minimal coverage of the cognitive neurosciences.

If we count up the number of publications in just cognitive neuroscience, we do see an exponential trend. Indeed, the area under the curve between 2000 to present day is substantially larger than before 2000, even for normalized publication counts. However, I'll leave it up to you to decide re: whether publication count would be a good proxy for the state of our knowledge.
(I personally think yes and no, but that requires a more thorough answer for another time... For all intents and purposes, it’s fine as a practical “guestimate”)

[Second panel plots the publication counts normalized by the absolute total number of publications indexed in PubMed that year x 100000]