Has anyone who knew what they were talking about every seriously suggested that your brain could "fill up", like a hard disk running out of free space? It seems like the closest you could get to that is your neurons becoming completely static (i.e. not-plastic), unable to form new connections or sever existing ones. That would kill you pretty quickly, or maybe another way to look at it is that it happens to you when you die, so the answer I guess could be "sort of, when you die".
But while you're alive? Assuming you don't just "add more space", to continue the analogy, it seems like what your brain does is probably closer to lossy compression, over and over again. And the important bits remain more-or-less readable.
To be technical ... from a information theoretical point of view your brain definitely can't hold infinite information so it definitely fills up in the sense you can't possibly be adding information without evicting/corrupting something else. I personally don't think we can reach that point within a human lifetime though.
The brain is constantly evicting, shifting, and changing memories as new stimuli come in and change the connectivity of our neurons. In that sense the brain, from inception, is full in the same sense that in an artificial neural network every neuron has a weight. The only thing that changes are the weights. (Note that I realize ANNs are a dramatically overly simplified representation of the brain, but the point remains).
The brain may also choose to re-encode a memory as part of a recognized pattern, in a sense, finding a way of compressing memories as recognizing them as part of a kind.
Maybe that excess capacity helps with efficiency? I know that some computer filesystems and flash devices get extremely slow when they start hitting 99% capacity.
It's like a piece of fabric one could keep weaving on - all the threads that get weaved on will always remain but only visible if it's near the top layer - That's my impression.
But while you're alive? Assuming you don't just "add more space", to continue the analogy, it seems like what your brain does is probably closer to lossy compression, over and over again. And the important bits remain more-or-less readable.