It's very likely that memory is "lossy" and holographic, such that you can keep adding more information indefinitely, but retain it with less and less accuracy.
Memory isn't a digital storage system with X gigabytes of capacity, and the inputs to memory aren't neat little packets. What we remember are a web of associations and patterns. Vastly oversimplified, we group similar memories together into a baseline, and remember the deviations from that.
If you walk the same route to work every day, you take in a huge amount of information the first few days: stretches of residential buildings, cars parked on the street, portions of curb painted red, trash in the gutters. In successive days different cars will come and go, front yards will get overgrown and trimmed, etc, but these details get mixed in with what you saw the first few days into a blurrier and blurrier sort of average. Trillions of impressions from hundreds of days take up very little more capacity than the first day did, and still tell you what you need to know about the route.
Imagine that there's a particular green car of common make and model, that was parked in front of a certain house about 3/4 of the time for the first hundred days you walked the route. Your memory reduces this to something like "green car usually along here somewhere". If the green car stops parking there, you'll probably notice "green car not there today" because that's a contrast to the common case that you've memorized. But if a week goes by with no sightings of the green car, it's very hard to remember which day you last saw it, because you don't remember individual sightings. (I actually had this happen to me with my own car; I discovered it stolen on a Wednesday and could not for the life of me recall whether I'd seen it on Monday or Tuesday even though I'd walked past the spot it should have been parked both days.)
As you remember more things, you remember them in less and less detail; you can't remember if you've been to New Orleans once or twice; you can't remember which memories were from your first trip to Las Vegas or your second, etc. So the brain never becomes full, it just becomes fuzzy.