Date: 201901
Local time of day: 1201-1800
Location: TEB.Airport, NJ
Flight conditions: VMC
Aircraft: Medium Transport; Low Wing; 2 Turbojet Eng
Operator: Air Taxi
Operating under: Part 135
Flight plan: IFR
Flight phase: Climb
Airspace: Class A ZDC
ATC / advisory: Center ZDC
Reporter organization: Air Taxi
Reporter function: Captain; Pilot Flying
Reporter qualification: Flight Crew Air Transport Pilot (ATP); Flight Crew Instrument; Flight Crew Multiengine
Human factors: Communication Breakdown
Anomaly: ATC Issue All Types; Conflict Airborne Conflict; Deviation - Altitude Overshoot; Deviation / Discrepancy - Procedural Clearance
Detector: Automation Aircraft RA; Person Flight Crew
When detected: In-flight
Result: Flight Crew Took Evasive Action
Contributing factors: Airspace Structure; Environment - Non Weather Related; Human Factors
Primary problem: Ambiguous

Narrative: TCAS RA event during climbout. No injuries; no damage; no emergency declared.Our flight was delayed on departure out of TEB due to ATC flow control restrictions over WHITE. Per TEB Ground control this was due to traffic saturation from the NYC airports going southbound. TEB was spacing departures at ten minute intervals. Our delay from passenger boarding to takeoff was approximately 1.5 hours. I mention this to note that traffic saturation along the route may have been a contributing factor to the IROP (Irregular Operations) event.IROP narrative begins between the CYN VOR and VILLS intersection on J209 in level flight at FL300. Center issues a clearance to climb to FL400. Clearance is read back verbatim and not challenged or corrected by ATC. Crew begins climbing to FL400. We are then handed off to Washington Center on 121.025 MHz. We check in with our current altitude of approximately FL320 and climbing to cleared altitude FL400. ATC responds acknowledging our check-in and does not correct or challenge the altitude clearance. Continuing our climb through approximately FL342 we receive a TA (traffic advisory) at approx XA55Z. This is unusual in a climb situation at high altitudes and immediately got our attention. Weather was night VMC on top of a lower cloud deck.As PF (pilot flying) I spotted the traffic quickly; Aircraft Y nearly head on with us at a high closure rate. Our climb rate at that point was approximately 1000 fpm. I immediately slowed the climb rate to zero / level flight realizing very quickly that the situation was not safe and we needed to act. At that point we received a TCAS RA (resolution advisory). The RA commanded a level descent of approximately 500 FPM. I disengaged the autopilot and hand-flew the RA. As the traffic passed overhead we were approximately 800 feet below Aircraft Y and descending per the RA. PM (Pilot Monitoring) immediately advised ATC that we were responding to an RA.Once the RA was over and we were clear of the traffic we advised ATC. PM advised ATC that we were continuing our climb per our previous clearance to FL400. ATC then asked if our altitude clearance was FL340. PM responded 'Negative; FL400.' ATC then stated that we were cleared to FL340. PM read back that clearance and we maintained FL340. No further traffic issues after this.At that point the other airplane came on the radio and asked ATC for our callsign; which was given. Other aircraft was Aircraft Y headed northbound level at FL350. He stated they received a TA during the event; but no RA.The RA maneuver commanded a slight descent of 500 FPM. The passengers were not aware of the RA and there were no negative Gs. The maneuver was smooth with no noticeable change from a passenger perspective. Entire event lasted less than one minute. In hindsight; it was strange that we were handed off several times quickly before reaching Washington Center after receiving a climb to FL400 clearance. There is a possibility that we took a handoff for a similar sounding callsign. Another possibility is that ATC issued us the clearance that was meant for the other aircraft with similar callsign. It is further possible that the handoff was also in error and/or ATC did not catch it when we took the handoff. ATC vigilance was needed in this situation and was missing. Crew vigilance also needed due to heavy traffic. Crew vigilance was compromised due to fatigue; distraction with a minor mx issue; and lack of nutrition due to long day with mx issues and no time to eat lunch or dinner. The clearance to climb to FL400 was unusual given the location; and could have been clarified/questioned earlier by crew. However; it was not completely out of the ordinary or unexpected given the 10 minute traffic spacing. These issues did not rise up to conscious concerns until after-the-fact.Suggestion for improvement is to clarify altitude clearances in the NYC area if anything doesn't seem correct; even for just a gut feeling.




Synopsis: Air taxi Captain reported an airborne conflict during climb due to a communication breakdown with ATC.