The Wikipedia page for Maxwell-Boltzmann statistics states:
In statistical mechanics, Maxwell–Boltzmann statistics describes the average distribution of non-interacting material particles over various energy states in thermal equilibrium and is applicable when the temperature is high enough or the particle density is low enough to render quantum effects negligible.
Here, it mentions that it is applicable when high temperatures are involved. This seems confusing to me since as far as I know the derivation for Boltzmann statistics only requires the following two assumptions:
The particles are distinguishable
The particles are weakly-interacting
The first assumption plays its role while we count the microstates and the second assumption shows up in the fact that the total energy is considered as sum of individual internal energies without any cross-terms. This is enough for deriving the Boltzmann exponential dependency and the constraints give the Lagrange multiplier values.
Now then why is it said on Wikipedia that high temperatures are required?
As a specific example, for a two energy level system of N particles the equation for specific heat is as follows: $$C=\frac {Nk_b(\theta/T)^2exp(-\theta/T)}{[1+exp(-\theta/T)]^2}$$ now is this expression wrong for $T$ tending to zero?