Abstract Instrumental variables (IVs) are extensively used to handle unmeasured confounding. However, weak IVs may cause problems. Many matched studies have considered strengthening an IV through discarding some of the sample. It is widely accepted that strengthening an IV tends to increase the power of non-parametric tests and sensitivity analyses. We re-evaluate this conventional wisdom and offer new insights. First, we evaluate the trade-off between IV strength and sample size assuming a valid IV and exhibit conditions under which strengthening an IV increases power. Second, we derive a criterion for checking the validity of a sensitivity analysis model with a continuous dose and show that the widely used Γ sensitivity analysis model, which was used to argue that strengthening an IV increases the power of sensitivity analyses in large samples, does not work for continuous IVs. Third, we quantify the bias of the Wald estimator with a possibly invalid IV and leverage it to develop a valid sensitivity analysis framework and show that strengthening an IV may or may not increase the power of sensitivity analyses. We use our framework to study the effect on premature babies of being delivered in a high technology/high volume neonatal intensive care unit.