Flight Crew Multiengine; Flight Crew Air Transport Pilot (ATP); Flight Crew Instrument
Situational Awareness; Communication Breakdown
ATC Issue All Types; Conflict Airborne Conflict; Conflict NMAC; Deviation - Altitude Excursion From Assigned Altitude
Flight Crew Took Evasive Action; Flight Crew FLC complied w / Automation / Advisory; Flight Crew Requested ATC Assistance / Clarification
Procedure; Human Factors; Environment - Non Weather Related
We were cleared to join the transition to the TEJAS4 STAR into IAH; and issued a clearance to climb from 11;000 ft to 17;000 ft. The Autopilot was on. I heard [Houston] Center give a traffic report to a [military] trainer; an '[Aircraft] out of 11 for 17' which I assumed was us. Center did not advise us of the traffic; but I was looking for it and I saw the traffic on the TCAS; descending and converging. Just before 12;000 ft. we had a TCAS TA then RA; at one point commanding vertical speed in excess of 3;000 fpm. The TCAS indicator was a red square; well inside the 3 mile ring on the TCAS screen and converging to our location. It appeared the [military trainer] was descending right on top of us; much like the TCAS avoidance exercise in recurrent training. I immediately disconnected the autopilot and pushed the nose down; taking evasive action as the TCAS commanded a high rate of descent to avoid collision. We announced our TCAS descent to Houston Center. After the event was over; we continued our climb to 17;000 ft. I queried the controller regarding the event. The [military trainer] pilot stated he had visual with us the entire time which I find dubious as we were far too close. At one point the TCAS indicated 300 ft. vertical separation.  We did not have a visual with the [military trainer]; largely due to the fact that he was approaching from our right side and was obscured by the rising sun. The Flight Attendant confirmed no injuries to passengers or herself and everyone was seated. I am unsure if the [military trainer] is equipped with TCAS; as the aircraft made no observable attempt to climb as would be expected when both aircraft are equipped with TCAS.Both ATC and the other pilot made critical errors in separation. ATC cleared the [military trainer] to descend too close to us and through our altitude; and the [military trainer] pilot made no discernible effort to alter flight path despite maintaining visual with us.
Air carrier Captain reported an NMAC during departure from IAH.