How do we know certain aspects of QM are unknowable? In a course on modern physics we are beginning to get into probability amplitudes and the inherent unknown in some things, say the position of an electron in an orbital.
How do we know that these things can't be precisely figured out? 
Could it be the case that there is a phenomenon or relationship we just aren't aware of?
Things like the laws of thermodynamics and magnetic dipoles can't be mathematically derived, but are taken as ubiquitously true because no counterexample has ever been found. Are probability amplitudes the same?
 A: It isn't that the position of the electron is unknown, but rather that the electron simply doesn't have a position.
The idea that an object has a position is such an intuitive one that it's hard to believe this might not be true. But a position, in the sense of a precisely defined location where we can find an object, is a macroscopic concept that simply doesn't exist for quantum objects. An electron in a hydrogen atom is delocalised and does not have a position for us to know.
In the early days of QM it was suggested that parameters like position do really exist but that we can't know them. This was known as the hidden variables theory. However as Robin suggests in a comment a theorem due to John Stewart Bell can be used to test whether hidden variables are consistent with observation, and to date the results suggest that the hidden variable theory is not consistent with experiment.
To address your last paragraph: quantum mechanics is a mathematical model that seems to describe the real world very well. Extraordinarily well in fact. If you're asking whether probability distributions really describe particles then you'll have to ask a philosopher what the word really means. All we know is that the predictions we make using quantum mechanics match experiment.
A: One interesting aspect related to this question is why we want to talk about a position and a momentum of an electron at all (for relevant phenomena where quantum effects become important). One motivation for the Copenhagen interpretation is that we are ultimately interested in measurement results, and those results (or at least their permanent record) have to be described in classical terms. This also implies that we have to introduce a Heisenberg cut somewhere between the quantum domain (where the phenomena happens) and the classical domain (where the measurements is registered).
You could try to avoid the need for a Heisenberg cut by taking it outside of the physical domain and inside the consciousness of the observer, but this probably misses the practically relevant points. Better use the Heisenberg cut to keep the domain which must be treated quantum mechanically small.

Let me try add an example based on a practical simulation perspective. Say I want to simulate the image formation in a low voltage scanning electron microscope by classically tracking incoming electrons (and generated secondary electrons) as they cross the specimen, but treat their interactions with the matter of the specimen quantum mechanically. Let's focus now on a scattering event with an inner shell electron. The position of the inner shell electron is relatively well known, since it is close to the nucleus. Consequently its momentum is less well know, i.e. its distribution is broad. But its momentum influences both the new momentum and direction in the incoming electron, and also the momentum and direction of the generated secondary electron. And the further tracking of both the incoming and the secondary electron now happens classically in the sense that both their position and momentum are assumed to be known. (Notice that the inner shell electron got assigned a classical momentum and position after the quantum mechanical interaction.)
You might protest that the Heisenberg cut was too early, and that the electrons too should be tracked quantum mechanically too between the interaction events. But this would be misguided, because the accuracy of the simulation is limited by other more important factors.
