In this paper, a single user channel is considered with an energy harvesting transmitter that receives its energy and data intermittently. The transmitter is equipped with a finite battery as well as a finite data buffer. The throughput maximization problem with a deadline is solved and the optimal transmission policy is obtained. The optimization problem is shown to yield a directional waterfilling solution with energy pumps. An alternative algorithmic solution is also presented that utilizes the recursive shortest path solution that was shown to be optimal for infinite data buffers in earlier work. Numerical results are provided to demonstrate the throughput performance of optimal policies as well as to assess the impact of the finite buffer on the throughput.