Air carrier with ZME initiated a descent responding to a clearance that was actually issued to another; same company aircraft; with similar four digit flight number.

Narrative

There were two aircraft with similar call signs on the same frequency with Memphis Center at the same time. We were Air Carrier XXXX; and the other one was Air Carrier XXXY. Climbing through FL270; ATC asked us to state our rate of climb. We responded 700FPM. Climbing through FL275; ATC said 'Air Carrier XXXX; expedite descent; descend and maintain FL240.' I looked at the TCAS; and saw traffic at our altitude 15 NM in front of us. First Officer read back the clearance; and we initiated an immediate decent (2500+ FPM) to FL240. Air Carrier XXXY did not query ATC on the clearance; and the controller did not challenge us when the clearance was read back. Descending through FL255; ATC stated that he showed us descending. We responded in the affirmative. He stated that the descent was for another aircraft; and we should still be climbing to FL330. I asked the controller specifically if any loss of separation had occurred; and he stated that it had not. A few minutes later; he cleared Air Carrier XXXX to FL180. Air Carrier XXXY questioned him; and he said that the FL180 clearance was for Air Carrier XXXX. Cause; similar call signs. We would have queried ATC about the descent if it had not been given as an expedited clearance with traffic in such close proximity. The inquiry about our rate of climb just moments prior also led me to believe that there was a possible traffic conflict.