Global sensitivity metrics are essential tools for assessing parameter importance in complex models, particularly when precise information about parameter values is unavailable. In many cases, such metrics are used to provide parameter rankings that allow for necessary dimension reduction in moderate-to-high dimensional systems. However, globally-derived sensitivity results may obscure localized variability in parameter sensitivities, resulting in misleading conclusions about parameter importance and ensuing consequences for subsequent tasks such as model calibration and surrogate model construction. In this study, we illustrate how discrepancies between globally- and locally-based sensitivity information may arise for an emerging sensitivity metric based on active subspace methodology, as well as for other commonly used sensitivity techniques. In response, we outline a framework that exploits the active subspace to evaluate the stability of parameter sensitivities over the admissible parameter space. This analysis allows one to determine the subregions of the parameter space in which a globally-derived sensitivity metric may be trustworthy. The proposed framework is illustrated on a collection of simple examples for ease of visualization, as well as the highly-applicable Lotka-Volterra model, for which we demonstrate how these issues may be exacerbated in higher dimensions. Our findings suggest that globally-derived sensitivity information should be treated with caution, and that incorporating analysis on local subregions may improve robustness and accuracy in downstream modeling tasks.