You seem to be interested in the case of mesoscopic systems. Here you will have to be precise about the preparation of the ensemble (and you will easily leave the realm of equilibrium thermodynamics if you're not careful).
Entropy is always the property of an ensemble not of a system. We get away with ignoring this in the macroscopic case because:
We look at the system at timescales large compared to the scale of time evolution of the internal degrees of freedom. So we can describe the systems by the time average and assume it is the same as the ensemble average (which holds for "reasonable" systems).
We look at very large systems, so we can average over small cells of the system, and so again describe our system as an ensemble with some average properties.
If you don't have this, e.g. because you have 1000 atoms in a harmonic trap, then, it's all about preparation. The questions are then ones like: How does the system interact with the environment (particle exchange, energy exchange)? Was the system prepared by taking part of a much larger system that was in equilibrium, and does it, thereby, "freeze" the equilibrium distribution of it?
If you do something like microcanonical preparation of the system (which you, of course, can't really do exactly experimentally), again and again – then you have a microcanonical ensemble, and this will have a well defined entropy, and this entropy will only take on certain discrete values. However, your system will no longer have a well defined temperature (because temperature can only be defined if entropy can be assumed to be continuous).
On the other hand, if you have a single two-level system connected to a heat bath (preparation in the canonical ensemble), and prepare this again and again, the entropy of your ensemble of two level systems is well defined again and continuously depends on the temperature of your heat bath. In this case, the system has the temperature of the heat bath by definition (!). But a single measurement of the two level system won't tell you the entropy – you have to measure again and again (as the entropy is a property of the ensemble not of the system).
As soon your system is small and there is no microcanonical preparation and no mechanism to equlibrate with a large system either – then equilibrium thermodynamics will totally break down. You can easily put five atoms in a trap with an arbitrary initial probability distribution, but you can't expect the laws of thermodynamics to hold then.