A growing literature uses value-added (VA) models to quantify principals' contributions to improving student outcomes. Principal VA is typically estimated using a connected networks model that includes both principal and school fixed effects (FE) to isolate principal effectiveness from fixed school factors that principals cannot control. While conceptually appealing, high-dimensional FE regression models require sufficient variation to produce accurate VA estimates. Using simulation methods applied to administrative data from Tennessee and New York City, we show that limited mobility of principals among schools yields connected networks that are extremely sparse, where VA estimates are either highly localized or statistically unreliable. Employing a random effects shrinkage estimator, however, can alleviate estimation error to increase the reliability of principal VA.