Date: 201509
Local time of day: 1801-2400
Location: ZZZ.Airport, US
Light: Night
Aircraft: Widebody; Low Wing; 2 Turbojet Eng
Operator: Air Carrier
Operating under: Part 121
Flight plan: IFR
Mission: Passenger
Flight phase: Taxi
Aircraft component: Nose Gear Tire
Component problem: Improperly Operated
Reporter organization: Air Carrier
Reporter function: Pilot Flying; Captain
Reporter qualification: Flight Crew Air Transport Pilot (ATP)
Human factors: Communication Breakdown; Training / Qualification; Situational Awareness
Second reporter function: First Officer; Pilot Not Flying
Anomaly: Aircraft Equipment Problem Less Severe; Deviation / Discrepancy - Procedural Published Material / Policy; Ground Event / Encounter Object
Detector: Person Flight Crew; Person Maintenance
When detected: Taxi
Result: Aircraft Aircraft Damaged; Flight Crew Took Evasive Action; General Maintenance Action
Contributing factors: Company Policy; Human Factors; Procedure
Primary problem: Procedure

Narrative: During the push back the ground crew cleared us to start all engines; so we started the right engine first. Upon completion of the push back we were instructed to set the brakes and responded and set the brake and started the left engine. Captain advised the ground crew that they were cleared to disconnect and to show the pin on the left side. On the after start flow we noticed that the left FWD fuel pump was illuminated with an EICAS message; so we told the ground crew on the headsets that we were working on an issue with the fuel pump so that they were aware of a possible delay in taxi but we did not tell them to stand by on headset as we knew we could be dispatch under the MEL if required. After the QRH procedure was complete successfully and the pump was functioning properly. A marshaller was standing to the left of the aircraft showing us the pin and giving the salute signal. We completed the after start check list and both pilots confirmed that the ramp was clear; all the marshallers were away from the aircraft and no one was standing in front performing the X signal. So; sure that the crew was disconnected; since salute was given and a pin was shown we started to move the aircraft; after just a couple of feet of aircraft movement we heard an unusual noise for the nose area; so we stopped immediately. The flight mechanic was sent to inspect and found no push tug; the pin had been removed and the tow bar was under the nose wheels. Both nose wheels were still inflated but had cuts on the side wall. The tow bar was showing some damage but the shear bolts were intact. The flight mechanic informed us that a both tires had to be replaced.After repairs were made we pushed back again.  During the push back the ground crew cleared us to start all engines; so we started the right engine first.  We were then advised that the push back was complete and on their command we set the brakes. As we are about to start the left engine  we notice on the side of the aircraft a marshaller showing us the pin and a salute; the captain disregarded the marshaller and informed the ground crew on the headset that it was not the time to show us the pin and giving a salute since the tug was will attached. Once the left engine start was complete we advised the ground crew that they were cleared to disconnect and now was the time to show the pin on the left side and give us the salute when clear of the aircraft. The pin was shown again; this time a marshaller was clearly visible with the X signal until all the equipment was removed. The marshaller removed the X signal and we taxied out.


Second narrative: During the delay; I was able to talk to someone I believe to be the ground manager. He informed me the tug had in fact pulled away; the tow bar was disconnected; but the tow bar had not been moved away.Ground crew should not show the pin and wave off the CA if in fact the aircraft is not cleared. This appears to be a training issue at this terminal as it happened during the second pushback


Synopsis: A B767 crew reported the Marshaller signaled clear to taxi; but after moving several feet the nose tires contacted the tow bar which was disconnected but not moved. After the nose tires were changed; the Marshaller's taxi clearance was again inappropriate because the equipment had not moved.