B737 Undifferentiated or Other Model
Flight Crew Air Transport Pilot (ATP); Flight Crew Multiengine; Flight Crew Instrument
Communication Breakdown; Time Pressure; Confusion; Situational Awareness
Conflict Ground Conflict; Critical; Deviation / Discrepancy - Procedural Published Material / Policy
Flight Crew Took Evasive Action
Human Factors
We were taxiing out of the north ramp area from Gate XX. We were cleared by ZZZ Ramp Control to taxi out. We contacted ZZZ Ground Control and the Controller gave us a taxi clearance to taxi to Runway XXL and to follow an [Aircraft Y] coming from our left on the taxiway. Both my FO (First Officer) and I understood this clearance and my FO read back the clearance correctly to the Ground Controller. Then a short time after that; the Ground Controller told an aircraft (that was taxiing behind the above mentioned Airbus) to give way to the [Aircraft X] (us). We don't know for sure if that [Aircraft Z] heard and/or acknowledged that clearance. And the Ground Controller was pretty busy. We're not exactly sure if Controller repeated the clearance either. We didn't hear a repeat. (I guess we'd have to check the tapes for that.)So as per our clearance; we started to taxi out taxiway after the [Aircraft Y] passed by us. The [Aircraft Z]; coming from our left (South); just kept taxiing northbound at high speed; with no signs of slowing down. My FO queried Ground to see if the [Aircraft Z] was going to give way to us. Ground asked [Aircraft Z] that and [Aircraft Z] replied 'Negative' and just kept going. He basically cut us off. I applied heavier than normal braking to stop and we kept a safe distance from this [Aircraft Z] while he sped past us. ZZZ Ground Control could have repeated the Clearance to the [Aircraft Z] and/or made sure that [Aircraft Z] understood the first 'give way to [Aircraft X]' clearance; but the Controller was very busy. And the [Aircraft Z] pilots should have heard the first clearance; but they also should have checked with Ground Control when they saw us coming out.
B737 Pilot had a ground conflict with another aircraft during taxi out for takeoff.