In this paper, we seek answer to the question: can a wireless sensing system with energy harvesting power supplies perform as well as one with conventional power supplies? Due to the stochastic nature of the energy harvested from the ambient environment, uniform sampling employed by conventional sensing systems is usually infeasible for energy harvesting sensing systems. We propose a simple best-effort sensing scheme, which defines a set of equally-spaced candidate sensing instants. At a given candidate sensing instant, the sensor will perform sensing if there is sufficient energy available, and it will remain silent otherwise. It is analytically shown that the percentage of silent candidate sensing instants diminishes as time increases, if and only if the average energy harvesting rate is no less than the average energy consumption rate. The theoretical results are then used to guide the design of a practical sensing system that monitors a time-varying event. Both analysis and simulations show that the energy harvesting system with the best-effort sensing scheme can asymptotically achieve the same mean squared error (MSE) performance as one with uniform sensing and deterministic energy sources. Therefore, we provide a positive answer to the question from both theoretical and practical aspects.