In this paper, we consider a wireless powered cooperative network, where two energy-constrained buffer-aided relays harvest energy from the source to assist the information delivery to the destination. To improve the spectral efficiency, a wireless powered successive relaying (WPSR) protocol is proposed to allow both relays to receive the information from the source and forward the buffered data to the destination, taking into consideration of the inter-relay-interference (IRI). We first reveal the impact of data buffers and energy storages at the relays by the achievable throughput comparison of the WPSR scheduling schemes with and without considering data buffers and energy storages. Our analysis validates the great potential of the buffer-and-forward data transmission and the harvest-store-use energy management strategies. Then, the network throughput is maximized via adaptive time and power allocation subject to the stability constraints of data buffers and energy buffers, and an adaptive wireless powered buffer-aided successive relaying (WPBSR) scheduling scheme is proposed. The proposed scheme approaches the optimal throughput of the wireless powered successive relaying network with bounded delay and finite length of data and energy buffers. Numerical results validate the analysis and noticeable spectral efficiency gain.