Directional antennas improve network performance by increasing the communication range and alleviating contention as proven, e.g., in cellular and ad-hoc networks. In principle, one may reap similar benefits in wireless sensor networks (WSNs), where energy concerns and reliability requirements make this antenna technology even more desirable. However, it is unclear how the shortcomings of directional antennas, e.g., increased likelihood of hidden terminals, affect WSNs. We quantitatively study these aspects for convergecast, a staple network functionality popular in WSN applications, e.g., for data collection. The integration of directional communication in convergecast protocols is non-trivial: probing wireless links between neighboring nodes is no longer feasible with single broadcast transmissions, as the antenna configuration depends on the target neighbor. This bears a great impact on the efficiency in building and maintaining the routing topology. We perform our study in simulation, based on an empirical model of an existing antenna prototype. This allows us to explore the parameter space efficiently yet realistically, a goal otherwise impossible without several antenna prototypes that, unlike WSN motes, are not readily available. Our results point to a negative answer, directional antennas, when used for WSN convergecast, provide limited benefits, appreciable only when certain specific conditions are met.