The answer is undoubtedly yes, because you didn't specify how small the population could be. Obviously, if you had a population of 2, then it's simply not possible for 2 people to know 99%+ of all modern knowledge.
But I'm guessing that what you really want is a self-sustaining civilization. In other words, you don't want the population to be so small that it goes extinct. If that's the case, then you want to learn about the concept of Minimum Viable Population: the smallest a population can be without a species going extinct over time.
One of the major factors is that you have to have enough DNA in the gene pool to not have inbreeding factors, over time, cause the population to become infertile. That bumps the number up to at least around 100, give or take.
But there are many other factors. The population has to be resilient to a disease or famine or natural disaster (etc) wiping out a chunk of them. There's debate as to what homo sapiens' true MVP is, but a good guess is that it's in the single-digit thousands.
So: could you have only a few thousand humans left and have mass knowledge loss?
Well, yes, depending on how you define "mass," but certainly a lot of knowledge would be lost, just from a statistical perspective alone this has to be the case:
- The population of earth right now is 7.9 billion. If 7,900 was your population size, that would mean you've lost 99.9999% of the population.
- Of any given profession, how many people would have to survive for knowledge of that profession to be passed down? Remember, it's not enough that some textbooks survived. Even assuming that your civilization is able to create printing presses, learning from unorganized, random books, without the aid of a teacher, is extremely time-consuming, and in general is not going to be a scalable way of people learning lost knowledge. Remember, your people are going to have to be spending the vast majority of their time just foraging for food, water, and protecting themselves and their food from the elements and animals, so people don't have all day to be reading books and trying to learn the hard way with no guidance (not to mention, most people don't have the proclivity for that anyway; remember only a small percentage of the population is nerds who enjoy that kind of thing).
- Of all the professions on earth, how many people have to survive in order for knowledge of their profession to get passed down? It will vary by profession. Carpentry? The answer may be as low as 5 (disclaimer: am not a carpenter). But for building computers? We're talking thousands, at a minimum. You would not believe how much specialized knowledge there is for manufacturing CPUs, let alone everything else involved. But suffice it to say, for some professions this number will be single digits, for others in the dozens, for others in the hundreds or thousands.
- And bear in mind that that many people need to be located together to share their combined knowledge. If it takes the knowledge of 2,000 people to build a CPU, it's not enough that 2,000 people survived...they have to be able to find each other and actually organize to do it.
- Ok so, given that number, what is the likelihood that that many people of that profession survived? For example, if it took 20 people who understand how to make an engine in order for that knowledge to be passed down, what are the odds that, when only .0000001% of the population survived the apocalypse, that that 7,900 people happened to include 20 people who knew how to build an engine? And those people all find each other and work together and have time for that?
And remember, it's not just the knowledge that you need, but also the whole supply chain and all of the resources and capabilities to extract those resources, everything that goes into making the thing, not just knowledge of how to make the thing if you were given all the materials on a silver platter.
And you have to pull off the organization of the whole thing.
Another way to look at it: you're one of the 7,900 people who survived the apocalypse. Let's say all 7,900 of you are located within a few hundred miles of each other. You're really outgoing, so you personally know 300 people. You have some knowledge of how to build an engine. You couldn't do it alone, but if you were able to get together with 20 other people who know about building engines, and you were able to convince the population to support you for many years, you could eventually start producing engines.
I say years because remember, you have to manufacture parts to make it, and that involves getting steel, and that involves forges...it wouldn't be enough for the 20 of you to be involved with trying to build engines, you would need hundreds of people at a minimum to be doing other tasks to provide you with the materials you need to build an engine.
Ok, let's say you get all of that buy-in. What are the odds that you even know 20 people, out of those 300 that you know, who have this kind of knowledge? Are 7% of your acquaintances experts in building engines? Not just that they know about it in theory, but they actually know actionable, specific details?
I think it's a good assumption that, given a population this small, the knowledge of how to build engines, as well as how to do a great many other things, would eventually die. Generating electricity is going to be hard, but possible in limited amounts. No one's going to make a full-blown power plant, but maybe some of the simpler ways of generating energy might happen in isolated areas. But without manufacturing happening, there's going to be a lot of difficulty using even that paucity of energy to make most items we're familiar with.
I think it's very plausible for you to get the end-state that you want.