Wouldn't that rather lead to forgetting things long gone rather than getting less sharp?
Also given that memories actually aren't stored as such, but recreated when remembered I am not sure how firm that cap is. But I am no expert on the subject so the cap might be very real.
Information in the brain is physically stored, and activated when needed. But not just your memories (episodic memory), but also language, motor skills, and recognition of sensory input. In my opinion, if you're immortal , one of two things will happen;
1) At some point, the brain starts overwriting connections to store new information. And not just in your memory, but everywhere new space is needed. Starting with the things that are used least often. Even in places like the cerebellum, where autonomous motor skills (walking, how to move your mouth when you want to speak certain words, everything you move automatically and ordinary mortals usually think they can never unlearn).
2) It takes a long while before things start getting overwritten, so it gets harder and harder to access information. You'd just be constantly lost in the endless networks your brain now contains. Your cognition would start to get slower and slower, because it would take longer and longer to use the part of the brain that's necessary.
It could also be somewhere in the middle, it depends on the physical limit to the brain's capacity to make new pathways. But yeah, either would be what TvTropes calls "The Fog of Ages", and I think either would be an interesting way to balance immortal characters!