  Estimating the parameters from $k$ independent Bin$(n,p)$ random variables,
when both parameters $n$ and $p$ are unknown, is relevant to a variety of
applications. It is particularly difficult if $n$ is large and $p$ is small.
Over the past decades, several articles have proposed Bayesian approaches to
estimate $n$ in this setting, but asymptotic results could only be established
recently in \cite{Schneider}. There, posterior contraction for $n$ is proven in
the problematic parameter regime where $n\rightarrow\infty$ and $p\rightarrow0$
at certain rates. In this article, we study numerically how far the theoretical
upper bound on $n$ can be relaxed in simulations without losing posterior
consistency.
