  For a model with many-to-one connectivity it is widely expected that
mean-field theory captures the exact many-particle $N\to\infty$ limit, and that
higher-order cumulant expansions of the Heisenberg equations converge to this
same limit whilst providing improved approximations at finite $N$. Here we show
that this is in fact not always the case. Instead, whether mean-field theory
correctly describes the large-$N$ limit depends on how the model parameters
scale with $N$, and the convergence of cumulant expansions may be non-uniform
across even and odd orders. Further, even when a higher-order cumulant
expansion does recover the correct limit, the error is not monotonic with $N$
and may exceed that of mean-field theory.
