The paper treats uplink scenario where M user equipments (UEs) send to a Base Station (BS), possibly via a common Relay Station (RS) that is equipped with a buffer. This is a multiple-access relay channel (MARC) aided by a buffer. We devise a protocol in which the transmission mode is selected adaptively, using the buffer at the RS in order to maximize the average system throughput. We consider the general case in which the RS and the BS can have limits on the maximal number of transmitters that can be received over the multiple access channel. In each slot there are three type possible actions: (A1) multiple UEs transmit at rates that enable BS to decode them (A2) multiple UEs transmit, the BS can only decode the messages partially, while the RS completely; (A3) RS forwards the side information to BS about the partially decoded messages, which are going to be combined and decoded entirely at the BS, while simultaneously a number of UEs sends new messages to the BS. The results show that the adaptive selection of direct and buffer-aided relay transmissions leads to significant average throughput gains.