How do we know neutrons and electrons are distinct particles on the same scale as protons? I'm aware my question may not even make sense when considering quantum physics, but please excuse my ignorance. We are taught in school that there are basically 3 particles that make up atoms (protons, neutrons, electrons) and that broadly for each 1 proton there will be 1 neutron and 1 electron. My question is how do we know that neutrons and electrons are not much smaller particles that just happen to lump together in predictable quantities due to their interactions with protons?
Let's say for instance that for each 1 proton you have 1000 neutrons and 1000 electrons. In this case, each neutron has 1/1000th the mass of a proton and each electron has 1/1000th the charge of a proton. It would seem to me that all the math still works just fine and our observations would still be the same.
We are also taught that you should think of an electron as something like a cloud rather than 1 distinct particle which is very counter-intuitive. In contrast, if 1 electron is really 1000 or a million, it is not just intuitive, but obvious that they would form as kind of cloud around the nucleus of an atom.
 A: 
My question is how do we know that neutrons and electrons are not much smaller particles that just happen to lump together in predictable quantities due to their interactions with protons?

Because we have done many experiments which confirm to us the size/mass of these particles. These experiments also tell us that neutrons and electrons have different charge (the neutron being electrically neutral).

In this case, each neutron has 1/1000th the mass of a proton and each electron has 1/1000th the charge of a proton. It would seem to me that all the math still works just fine and our observations would still be the same.

The observations and mathematics would be vastly different and not reflecting of reality.

We are also taught that you should think of an electron as something like a cloud rather than 1 distinct particle which is very counter-intuitive.

More exactly, a probability cloud.
It is not counterintuitive if you consider the quantum mechanical description of the electron (and all quantum particles). Quantum mechanics is an incredibly accurate description of nature.

In contrast, if 1 electron is really 1000 or a million, it is not just intuitive, but obvious that they would form as kind of cloud around the nucleus of an atom.

There is a huge difference between a probability cloud and this.
This is not intuitive given our understanding of quantum physics and observations/experiments.
If you had that many electrons confined to this region, Coulomb repulsion would probably push the electrons apart and it is not likely you will have a localised “cloud”.
You would at the very least need an identical amount of oppositely charged protons, but  that is not possible in this model you have proposed.
A: 
If the electron were made up of smaller particles then we would expect to see some evidence of these smaller particles (gandalf61)

Nice argument for the beginning. Because there is a phenomenon where exactly this happens. During the process of annihilation of an electron and a positron we mainly observe three or even more particles. These are photons.
Some aspects following from this phenomenon (focused on the electron, but also applicable to the proton and partly to the neutron):

*

*The electron generates respectively possesses an electric and a magnetic field. The photon is a propagation of an electric and a magnetic field. But it would be too brief to say that electrons consist of photons, as we observe them.

*The photon is an elementary particle in the standard model. To be precise, however, photons can be distinguished by their energy content. All with c in vacuum and with two oscillating field components they have very different momentum. So they are more a class of particles.

Maybe some day there will be a proposal to built up the two fields from more elementary particles. This could be only two particles, because both the electric field and the magnetic field are dipole fields (in the meaning that any separation of charges lead to two poles and any alignment of the electrons magnetic moments (a permanent magnet for example) is a dipole).
Up to this time a field is represented (modelled) by non real existing (as often emphasised) field lines and the exchange between the particle fields happens by virtual (but not real existing, as often emphasised) photons. To say it shorter, nobody cares about the nature (the inner structure) of these fields.

How do we know neutrons and electrons are distinct particles on the same scale as protons?

There is no doubt that all three particles are subatomic particles, and we distinguish them by chemical processes or electromagnetism or many other interesting things.
Understanding the field interaction between them is not the last word as long as we need explanations that include virtual particles. And if we take a look at the inner structure of the fields, we may also come to a deeper understanding of the inner structure of the electron and the other subatomic particles.
A: If the electron were made up of smaller particles then we would expect to see some evidence of these smaller particles in particle collisions observed at CERN and other particle collider. We have never seen any such evidence. So as far as we know the electron is an elementary particle.
In the case of the proton and the neutron we have seen smaller particles that make up protons and neutrons. These particles are called quarks and each proton and neutron contains three quarks. Individual quarks are very difficult to observe because they are bound together so strongly, but we definitely know that they exist. So we know that protons and neutrons are not elementary particles.
