Every patient I’ve had in the ER just slipped and fell on it. Crazy odds. Yeah, our bodies can’t push everything out whether because of size, shape, or integrity. I think the worst was the guy who had peeled ginger that got stuck. BDSM thing for him, used to be a form of torture. He was in a lot of pain and it took a while to get him to endoscopy.

Basically because it's not soft enough.

Your body "pushes" things out by squeezing in a "rolling" motion. Like running a rolling pin over a tube of toothpaste.

Picture each of those little segments contracting and relaxing in sequence to slowly move things along, until it gets dumped in the rectum, where it sits until you and it come to an understanding.
Bunch of muscles then move things around to get things lined up, since normally things rest in a way that helps keep things from just falling out. Anal sphincter also does this, but it's the difference between folding the chip bag closed, using a chip clip or both.
Once it's all lined up, it does that rolling squeeze again, takes off the chip clip and things proceed in a routine fashion.

So if instead of what it's used to, it's dealing with something like a cucumber, it can end up with the end up around that curve at the top of the rectum.
The tapered inside near the anal sphincter means that when your vegetable goes in, the muscle can squeeze against the end and make the situation more of a commitment than people had planned for.
Once there, it can run into a few more hurdles. The muscles near the top can't really do anything but squeeze the sides. If it's not squishy and there's no angle, it's not going to be able to do anything because it just doesn't have the angle. Even if there is an angle, like your cucumber didn't go all the way, it's going to be squeezing at an awkward angle to try to push something inflexible through the opening in the stronger anal sphincter.
Usually the softness lets things find a way with some mutual give and take, but even normally things can get a bit firm and get some resistance that can be uncomfortable to work through.

Turns out I think I remember more of my anatomy and physiology classes than I thought.

Well, when a man loves a mason jar very much…

Afaik, your intestines have an involuntary motion called peristalsis that keeps your poop moving. Sometimes, this can backfire with items that go the wrong way because they'll get sucked up. Your sphincter also tends to stay shut and wrap around irregularly shaped items. Your intestines aren't straight; they're muscles that bend around items, which can trap them. Some items, like bottles and jars, are concave enough to create a vacuum that keeps them in place. You'll have a hard time getting anything out that's not soft like feces unless the item has a wide enough base to stop it from traveling past your anus.

Disclaimer: I'm not a doctor and some surgeon will probably tell you more details I missed or got completely wrong.

Just a friendly reminder: flared bases are important

So this is quite the read, I recommend it:

https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3577617/

But it doesn’t answer the question. I’m, hnnnnnng working on it.

I'm guessing that at some point your rectum cramps closed and the foreign object ends up being trapped behind the sphincter.

Note that I'm not a medical professional, nor do I pretend to be one on the internet. I do have personal lived experience of constipation induced cramping and it's not fun, not to mention, painful to the point of passing out.

Someone should make a wikiHow with this title

I’ve worked in GI. You wouldn’t believe the things people put in their butt and then try to deny later. I’ve…seen…everything.