Date: 202102
Local time of day: 1201-1800
Location: ZZZ.ARTCC, US
Flight conditions: VMC
Light: Daylight
Aircraft: Medium Transport
Operator: Corporate
Operating under: Part 91
Flight plan: IFR
Mission: Passenger
Flight phase: Climb
Airspace: Class A ZZZ
ATC / advisory: Center ZZZ
Second aircraft: Medium Transport
Reporter organization: Corporate
Reporter function: Pilot Not Flying; Captain
Reporter qualification: Flight Crew Air Transport Pilot (ATP); Flight Crew Multiengine; Flight Crew Instrument; Flight Crew Flight Instructor
Human factors: Situational Awareness; Time Pressure; Communication Breakdown; Confusion
Anomaly: ATC Issue All Types; Conflict Airborne Conflict; Deviation - Altitude Excursion From Assigned Altitude; Deviation / Discrepancy - Procedural Published Material / Policy; Deviation / Discrepancy - Procedural Clearance
Detector: Automation Air Traffic Control; Automation Aircraft TA; Person Air Traffic Control
When detected: In-flight
Result: Air Traffic Control Issued Advisory / Alert; Flight Crew Became Reoriented
Contributing factors: Human Factors; Environment - Non Weather Related
Primary problem: Human Factors

Narrative: We were returning to ZZZ from ZZZ1 late on the afternoon. The enroute environment was apparently busier than the recent 'new normal' for that time of year as we had been limited in our climb clearances to between one- and three-thousand feet per climb clearance since our departure 25 minutes earlier.  Our most recent climb clearance was to FL400 (on our way to our filed FL410). Subsequently; ATC called traffic (I don't recall the model or the horizontal distance away) at '1;000 feet ABOVE YOUR ASSIGNED ALTITUDE' (emphasis mine). I would have replied 'Looking' or some similar acknowledgment. I did look and then subsequently noticed my FO (who was PF and in the left seat) had shallowed our climb to 100 FPM or so out of FL390 -- significantly less than the 5 or 600 FPM I would have expected. We were also very light; so a climb rate that shallow would have soon required a power reduction. At about the same time we got a TA and the Controller asked us to confirm we were level (or leveling?) at FL390.  I don't recall the exact exchange; but I would have replied something to the effect that we were climbing to FL400; he would have said something to the effect of maintain FL390.  At that point I fully processed what was happening; gave silent kudos to my FO for correctly interpreting the bad direction this was going; and directed him to 'push.'  As what we had was a TA; not an RA; the autopilot was left engaged; but he did direct the nose down and I dialed 390 into the ASEL.  The Controller asked us to confirm we were at FL390. I interpreted that as him wanting that confirmation on tape and I gave it to him; although we were; in fact; changing our vertical direction from up to down at FL393 at that point.We are a very structured flight department. In order for an airplane to be directed to an altitude during enroute climb or descent; the Controller has to assign that altitude; the PNF has to read it back and dial it into the ASEL; the PF has to verbally and independently repeat that altitude for both of us to hear; and has to acknowledge that he sees that same altitude displayed in front of him. Owing to that routine sequence and cadence of events; we are both 100% certain that we were assigned FL400.That said; I can allow as how I have possibly been wrong at least once in my flying career. My FO was hired on just as COVID came on the scene last year.  With the reduced amount of flying we've been doing since; he is still considered 'new' to the operation and as such not fully comfortable with our procedures. By that; I mean that I sometimes sense a hesitation for him to be as assertive as he should be and as assertive as he would be by now had this been a normal year. That notwithstanding; I would expect any pilot in our organization to question; in no uncertain terms; anything they were uncertain about; especially an ATC clearance. Likewise; no one in our organization; from Director of Flight Ops down; would hesitate to ascertain verification of any such uncertainty; no matter who questioned it. That's the way we move airplanes without bending them. I'm still 100% certain we were cleared to FL400.In retrospect; and we observed this at the time; the primary thing that would have alerted us to this unfolding conflict would have been for the Controller to have called the traffic as '...opposite direction at FL400.' That would have raised the hackles on the back of my neck and elicited from me a request to confirm our assigned altitude. There is no error-trapping possible when traffic is called as '1;000 feet above your assigned altitude.'Additionally; and this is not a negative; just observation and fact: we're flying the oldest; and by far the most reliable; airplane in our fleet of four; however; owing to her age; we're looking at a pair of >XX-year-old TCAS displays on dedicated three-inch round screens. There are no course overlays; no distance rings and; especially on the longer ranges (20 and 40 mile); no way to preciselyassess the relative position; direction or closure rate of traffic approaching; especially when doing so nearly head on. Technology is good. Newer technology can sometimes be better.On a positive note; we did visually acquire the traffic while far enough apart that had evasive action been necessary (it wasn't); we would have had plenty of time to take it. We did debrief this incident immediately after its occurrence; replayed the events out loud and decided the error was ultimately ATC's; that we had done everything we could to mitigate the repercussions to the controller and that we were not at fault.




Synopsis: Corporate Captain reported an altitude deviation that caused an airborne conflict (TA) and cites Controller/Pilot communication and the First Officer's (PF) lack of flying as contributing factors.