The authors discuss effective well-conditioning of the linear system \(Ax=b\). The condition number \(K(A)=\| A\| \| A^{-1}\|\) is often an overly conservative measure of the sensitivity of x under perturbations of \(\Delta\) A and \(\Delta\) b to A and b respectively. Two practical cases in which the sensitivity of x may be significantly less than the worst case predicted by K(A) are presented. The first characterizes a class of Vandermonde matrices and right-hand-sides and the second a FFT-based fast Poisson solver, for each of which accurate solutions may be obtained. For Vandermonde systems \textit{N. J. Higham} [Numer. Math. 50, 613-532 (1987; Zbl 0595.65029)] has shown that the algorithm of \textit{A. Björck} and \textit{V. Pereyra} [Math. Comput. 24, 893-903 (1971; Zbl 0221.65054)] gives relative errors in the non-zero components of x which are independent of K(A) provided that scalars \(\alpha_ j\) of the Vandermonde matrix are in ascending order and that the elements of the right-hand side b oscillate in sign. The computation of accurate solutions to the discretized one-dimensional Poisson problem is discussed and the authors' experiments indicate that a fast Poisson solver composed entirely of fast sine transforms will have better numerical performance than the more common fast transform with tridiagonal solving.