Many studies have considered building a stronger instrumental variable (IV) from the original, possibly weak, IV in a matched study at the cost of not using some of the samples. It is widely accepted that strengthening an IV may render nonparametric tests more powerful and typically increases the power of sensitivity analyses. In this work, we show that this conventional wisdom might be wrong. We consider matched observational studies from three perspectives: (1) We evaluate the trade-off between IV strength and sample size on nonparametric tests assuming the IV is valid and show that when strengthening an IV increases power and when does not. (2) We derive a necessary condition for a valid sensitivity analysis model with continuous doses. We show that the Gamma sensitivity analysis model, which has been previously used to come to the conclusion that strengthening an IV makes studies less sensitive to bias, does not apply to the continuous IV setting and thus this previously reached conclusion may be invalid. (3) We develop a valid sensitivity analysis framework and use it to show that strengthening an IV may increase or decrease the power of sensitivity analyses.