Can a Bose gas be saturated?

In Einstein's textbook picture of an ideal gas, Bose-Einstein condensation is driven by purely statistical saturation of the excited states of the system. Experiments on dilute ultracold atomic gases are celebrated as realizations of Bose-Einstein condensation in close to its purely statistical form. Here we scrutinise this point of view using an ultracold gas of potassium ($^{39}$K) atoms, in which the strength of interactions can be tuned via a Feshbach scattering resonance. We first show that under typical experimental conditions a partially condensed atomic gas strongly deviates from the textbook concept of a saturated vapour. We then use measurements at a range of interaction strengths and temperatures to extrapolate to the non-interacting limit, and prove that in this limit the behaviour of a Bose gas is consistent with the saturation picture.