  In the recent years there has been an increased interest in studying
regularity properties of the derivatives of stochastic evolution equations
(SEEs) with respect to their initial values. In particular, in the scientific
literature it has been shown for every natural number $n\in\mathbb{N}$ that if
the nonlinear drift coefficient and the nonlinear diffusion coefficient of the
considered SEE are $n$-times continuously Fr\'{e}chet differentiable, then the
solution of the considered SEE is also $n$-times continuously Fr\'{e}chet
differentiable with respect to its initial value and the corresponding
derivative processes satisfy a suitable regularity property in the sense that
the $n$-th derivative process can be extended continuously to $n$-linear
operators on negative Sobolev-type spaces with regularity parameters
$\delta_1,\delta_2,\ldots,\delta_n\in[0,1/2)$ provided that the condition $
\sum^n_{i=1} \delta_i < 1/2 $ is satisfied. The main contribution of this paper
is to reveal that this condition can essentially not be relaxed.
