  Both practice and analysis of adaptive $p$-FEMs and $hp$-FEMs raise the
question what increment in the current polynomial degree $p$ guarantees a
$p$-independent reduction of the Galerkin error. We answer this question for
the $p$-FEM in the simplified context of homogeneous Dirichlet problems for the
Poisson equation in the two dimensional unit square with polynomial data of
degree $p$. We show that an increment proportional to $p$ yields a $p$-robust
error reduction and provide computational evidence that a constant increment
does not.
