Air carrier Captain reported a near-collision during taxi at DCA airport.

Narrative

Upon landing we were told to taxi on Runway 33 and enter the hold bay for 15. We parked at the north end of the hold bay and waited for our gate to open. While we were waiting 2 other Company aircraft parked in the hold bay to our right. A 3rd aircraft exited Runway 33 and came to a stop on taxiway N; blocking all aircraft in the hold bay. Ground then cleared the plane on N to taxi; giving way to aircraft pushing. Ground then instructed us to follow company out of the block; taxi N short of the alley. The aircraft on N began to taxi. I turned on the taxi light and the FO and I cleared our sides prior to movement. As we pulled onto N the FO said the aircraft off our right wingtip in the hold bay was moving. I looked over and saw the airplane moving. The crew slammed on their brakes and came to a stop. We stopped and verified we still had wingtip clearance. I asked the FO to get clarification from Ground about our taxi. The Ground frequency was very congested. Before we could ask; Ground told the aircraft ahead and us to continue taxi then told us to hold. Ground then cleared us into gate.