  Motivated by the increasing popularity and the seemingly broad applicability
of pair-copula constructions underlined by numerous publications in the last
decade, in this contribution we tackle the unavoidable question on how flexible
and simplifying the commonly used `simplifying assumption' is from an analytic
perspective and provide answers to two related open questions posed by Nagler
and Czado in 2016. Aiming at a simplest possible setup for deriving the main
results we first focus on the three-dimensional setting. We prove that the
family of simplified copulas is flexible in the sense that it is dense in the
set of all three-dimensional co\-pulas with respect to the uniform metric
$d_\infty$ - considering stronger notions of convergence like the one induced
by the metric $D_1$, by weak conditional convergence, by total variation, or by
Kullback-Leibler divergence, however, the family even turn out to be nowhere
dense and hence insufficient for any kind of flexible approximation.
Furthermore, returning to $d_\infty$ we show that the partial vine copula is
never the optimal simplified copula approximation of a given, non-simplified
copula $C$, and derive examples illustrating that the corresponding
approximation error can be strikingly large and extend to more than 28\% of the
diameter of the metric space. Moreover, the mapping $\psi$ assigning each
three-dimensional copula its unique partial vine copula turns out to be
discontinuous with respect to $d_\infty$ (but continuous with respect to $D_1$
and to weak conditional convergence), implying a surprising sensitivity of
partial vine copula approximations. The afore-mentioned main results concerning
$d_\infty$ are then extended to the general multivariate setting.
