The setting that I am working on has a population of humans living in a place that does not have abundant copper or tin resources, so it is unlikely that they'd discover Bronze. However, bog iron is reasonably common, so is it feasible that such a civilisation might make the jump from pottery kilns directly to iron bloomery?
I imagine they could. Many people make the mistake of under-estimating both the intelligence and curiosity of ancient peoples; while it is clear from the record they did NOT always follow a logical path, they did many things just because they had the time and they were curious to see what would happen. Especially when they weren't that hungry, and just fooling around. Where is the logical path to the atlatl; if it wasn't just fooling around with sticks, or a genius insight that resulted in fooling around with sticks? How in the world did they invent the boomerang and become accurate with it? Or the sling, if you ever try to learn that on your own, start far from anything breakable.
When food is plentiful, hunter gatherers have time on their hands to do things. Without food preservation it doesn't do any good to hunt for weeks in advance. So they mess with things. Trying to cook different plants, make weapons, etc. One of the things they can mess with is seeing how hot a fire can get.
Crucible steel was being made in 800 AD, and can be made in clay ovens if you have the right mud. This melts iron and other ingredients in a clay crucible inside a clay oven. If you have bog iron, somebody somewhere may have thought they could purify it by fire, and just fooled around with making the fire hotter and hotter. It could be a hobby for years.
I don't think you need anything else to realize you can actually melt it; the objective could just be whim: How hot can you get it? Not much different from how far can you sling it or throw it.
I certainly don't think it is impossible. Ancient people built thousands of stone monuments and invented all the mechanics needed to dig out, shape and transport stones weighing tens of tons, hundreds of miles. Why? Because they could, and they had some idea in mind, be it religion or practical astronomy or both.
Fire has certainly been an object of reverence too, no reason to think early people would not go to great lengths in their exploration of it, and their exploration of things that can survive it or be transformed by it.
Homo Sapiens Sapiens has been around for at least 60,000 years and there are clear hints we've been around 250,000 years. That is us, human beings of our same exact intelligence and curiosity and reasoning capability. Obviously without the accumulation of our education and knowledge, but with language, skills, and both the physical and mental dexterity we have.
Far too often, we like to pretend these people were idiots or infants or dullards that needed to be led on a very simple path of technological discovery. I don't think so. I think fire was a very big part of their lives and they would have naturally explored all kinds of aspects of it, and could quite naturally have discovered the smelting of iron, not by knowing it would happen, but just by wondering what would happen if they got the oven hot enough. And, I will point out, the oven doesn't have to get very hot before the metal begins to soften and can be hammered by rocks.
These aren't dummies we are talking about, they are us, with just as many insightful geniuses as us. They had fewer "shoulders of giants" to stand upon, but that doesn't mean they didn't have a few giant intellects and insights of their own, for the problems that faced them.
You really need at least copper, copper can be smelted in a pottery kiln. Pottery kilns (760 C) don't get anywhere near the smelting or even forging temprature of iron (1500 C). only purpose built furnaces designed to achieve very high temperatures (forced air, charcoal, ext) will smelt iron, you can develop those progressively from other metals but not straight from a pottery kiln.
You also really need copper to realize smelting is a thing. Copper predates other tool metals and even bronze by a large margin (~6000 years) and it takes several thousand years of using copper before people learn they can extract copper from ores through heating. So people had to have copper around in tools for thousands of years before they even figured out they could smelt copper out of rocks, which is far easier than iron even though iron ore is far easier to come by.
The proof is in the cultures that didn't discover smelting. Metallic meteorite iron has been used without other metal technology (Cape York Inuit). However, said cultures never developed smelting technology there is no reason for them to believe they can heat up rocks and extract metal, and these are people who know you can heat iron to make it softer. Your people will not develop bloomery from nothing they need to understand useful metals can be extracted from rock by heating it to melting. In our history that has only been copper, other metals are either far too rare or even harder to refine than iron. Hundreds of cultures never develop metallurgy they were not less intelligence and its not due to lack of iron iron ore is nearly everywhere (excluding a few small islands). Usually it is because they did not have access to native copper, which is far rarer. Technology just requires certain resources and techniques to build on, to progress, without them the chance of stumbling across the right process by accident is just too unlikely.
The progression is pretty predictable and occurred multiple times, early people use pre existing metallic state metals like copper, then someone discovers they can heat certain rocks and get copper, which was very valuable, then they start testing every rock they can find and experimenting with different furnaces, higher and higher heats, fluxes, refining and other methods to extract better qualities or other metals.
Bog iron is not good for learning to work iron, bog iron is a easy (relatively) way to collect iron ore once you know about it but you are not getting much iron per amount of bog iron. The first smelting likely started with much higher quality ores like hematite which don't require processing. culture switch to bog iron once they have harvested out all the better iron ores.
By way of gold.
Even if your people don't have copper, they might have gold. Gold nuggets occur naturally in streams and lowlands. The melting point is similar to that of copper. No-one is going to make a weapon out of gold because it is so soft, but the main use for native copper was not weapons, but jewelry. If your people made decorative items out of hammered gold nuggets and then figured out they could melt the gold in a ceramic forge, they might figure the same thing out about the bog iron.
Or they might not. The Aztecs had the tech to melt copper and gold but not iron. The history of iron would make good reading. My suspicion is that it is one of those inventions, like the alphabet, that happened one time then spread everywhere it could spread.
I think historical interest in bog iron is from people who knew about iron and were looking for it. Unlike a gold nugget, I am not sure the native lumps of bog iron are good for much as is. If this is going to be part of your story, you might have someone make a knife from a meteor and then as it rusts, perceive its similarity to the bog iron lumps he uses for sling shots and fishing weights. A story getting into the weeds of tech development like that will never be made into a movie but I would like to read it!
It's possible because it happened in our history: some peoples in Africa didn't go through the Bronze Age but had an Iron Age at similar dates than those lands who had bronze: https://en.m.wikipedia.org/wiki/Iron_Age#Sub-Saharan_Africa
bronze age is, in fact a misconception.
in fact, only the Mediterraneans and the chinese have went through the bronze age, and both cultures started widespread use bronze after their surrounding culture that have started smelting iron.
it turns out, iron and it's smelting started from meteoric iron, then worked through the forge into bloomery iron. Meteoric iron CAN be worked cold, and the first iron users realizes that heating the Meteor makes it easier to work with.
bog iron started as a pottery pigment, and making glazed ceramics needs high temperatures: temperatures high enough for bloomery smelting. iron comes as a hard, dense lump, ductile when removed, which prompts any one who first discovered it think it as a good tool material, before discovering forging almost immediately.