Mixtures of 1-alkanols are a textbook example of the concept of ideal mixtures. Yet, such mixtures have a very strong local order due to the hydrogen bonding interactions, with a strong tendency for chain formation. Despite this apparent non-ideality, the Kirkwood-Buff integrals of such system exhibit near ideal behavior. This dual property can be used to test the calculations of the Kirkwood-Buff integrals in a controlled mixing situation, and clarify many points, in particular the statistical problems that can be encountered. By studying the methanol-ethanol mixtures, we uncover an interesting physical asymmetry between low methanol and low ethanol concentrations, which can produce statistical artifacts in the calculation of Kirkwood-Buff integrals, illustrating and exemplifying some of the difficulties encountered in such calculations. Finally, liquid state integral equations results for these mixtures are reported. They help demonstrate that thermodynamic ideality hides complex correlations and microscopic non-ideality.