PO boxes can be tiny. Where do they put the mail if it is full?!
At the PO where I have my box, if your box gets packed up too much, they will move your mail to a bank of about 20 *larger boxes, and they will leave a key in your box with a number corresponding to a box in the the “overflow” box bank. You then unlock that box, get your mail and leave the key.
You don’t have a choice to leave the key, because once you unlock the box, that also locks the key in the lock, and it takes another key by a postal workers to unlock that key.
*I’ll check on the size later, but they are roughly about 12" x 12" x 12".
In the 90s at a small town Post Office, they kept overflow rubber banded in the back, and put a small card in the box indicating that you had a package or overflow and needed to come to the window.
My PO box is similar to Jacob’s. They leave me a note saying my box is full and to come to the window with a suggestion that I upgrade to a bigger box. Which is not needed, I just need to get my lazy butt to the PO more often 
They’ll put the mail somewhere, notify you any way they can, and threaten to take your box away if you don’t come in and pick up your mail more often. I don’t know if they can back up the threats.
This is how the cluster boxes in Canada work too, either for overflow or large packages (many areas do not have door-to-door delivery - just neighbourhood cluster boxes). Except I’ve never seen one where the key gets locked into the door - it works on the honour system here where people are expected to just drop the key back into the outgoing mail slot!
This is usually the way it works in Apartment complexes in the US too. We have our standard mail boxes and then below it there are two larger boxes with keys in them. You just leave the key in the door once you collect your mail/packages. But if the boxes are full already, or you just have too much mail, you’ll get a note saying you have to come to the post office to pick it up.
Do they throw it away if you don’t respond? Eventually, I mean?
I have a business with a PO Box, and if I’m on vacation, and a large flux of mail came that week, the Post Office would put the overflow into a carry bin and leave a note in my PO Box to come to the desk to get that carry bin…not sure how long they are obligated to keep this up though.
You get a card, telling you to come to the counter to get your mail. Note: you can ONLY get the mail when the counter is open, typically 8-5 weekdays. (Our PO is so tiny, they close for lunch)
In bigger cities, you need to bring the card and some sort of identification to pick up the mail. At our PO, if the clerk sees you drive up, he usually has the mail waiting for you when you come inside.
~VOW
I have a USPS (US) PO box, and when I signed up for it, I was offered several sizes. Occasionally, I receive something that is too big to fit, and I either get a key to one of the big lockers in the facility or else I receive a slip of paper telling me to see a clerk, who gives me the item. I don’t think I’ve ever had to deal with a completely full box, though. It’s been stuffed, but there was still the ability to slip another envelope or card in. I’m just speculating here, but I’d guess that if I consistently received many many oversized items and/or consistently let my box fill to the brim, they might ask me to upgrade to a larger box.
A friend of mine has a PO box down here, but spends most of his time up north.
When he comes down to visit, he picks up his mail, usually in several plastic carry bins.
I think I have two in my garage right now, filled with magazines for him, from the last time he visited.
-D/a
This, of course, is also how they deliver packages to your P. O. Box that are too big to fit in the box. It’s a standard thing I’ve seen at post offices wherever they have P. O. Boxes.
ETA: In light of post #3 above, I should clarify that they do this at all larger post offices I’ve seen.
No, it gets returned to sender. First class, that is. 2nd class (magazines, newspapers) I think get tossed & the sender just gets a notice. 3rd class (bulk) gets tossed, and the sender only gets the notice if they requested (“Address Service Requested” printed on the piece), and they have to pay for each of those.
For a while, I had a PO box just to pick up packages at a more convenient location (the PO for my ZIP code was in a, um, bad part of town, and my apartment building wouldn’t hold on to packages). For a while I didn’t have any packages delivered so I didn’t check the PO box, and I got a call saying it was full of spam mail.
I’ve had one or more USPS PO Boxes for at least the past 20 years - all in the New Brunswick, NJ area. I always get a card if there is too much mail to fit. But I have to say, it is truly amazing how much mail they can stuff into a small box. Sometimes it’s jammed in there in such a way that you can almost feel the frustration of the poor clerk delivering the mail. It virtually screams ‘where the **** are you and why the **** haven’t you picked up your damned mail.’
I like the idea of getting a key to a larger box though. That would make it a little easier to get your excess mail since box lobbies tend to be open slightly long than the counter lobby. Plus you don’t have to bother with showing ID.