Date: 202006
Local time of day: 1801-2400
Location: ZZZ.Airport, US
Light: Daylight
Aircraft: Commercial Fixed Wing
Operator: Air Carrier
Operating under: Part 121
Flight plan: IFR
Mission: Passenger
Flight phase: Descent
Airspace: Class B ZZZ
ATC / advisory: TRACON ZZZ
Reporter organization: Air Carrier
Reporter function: Pilot Flying; First Officer
Reporter qualification: Flight Crew Air Transport Pilot (ATP); Flight Crew Instrument; Flight Crew Multiengine
Human factors: Confusion; Communication Breakdown
Second reporter function: Pilot Not Flying; Captain
Anomaly: ATC Issue All Types; Deviation - Altitude Excursion From Assigned Altitude; Deviation / Discrepancy - Procedural Clearance
Detector: Person Flight Crew
When detected: In-flight
Result: Flight Crew Returned To Clearance
Contributing factors: Environment - Non Weather Related; Human Factors
Primary problem: Human Factors

Narrative: On the ZZZZZ Arrival we were at 6;000 feet and the radios were busy. There were numerous aircraft with similar call signs. We were given clearance off the STAR and cleared direct ZZZZZ1 for ILS XR. We both thought the Controller also cleared us to 4;000 feet. I asked the Capitan; PM; if we were cleared to 4;000 feet; and he said; 'Yes.' I also thought I heard the Controller clear us to 4;000 feet. I started down in Vertical Speed mode 1;000 FPM; and when we approached 5;000 feet; ATC queried why we were descending; and to just level off at 5;000 feet. The PM told ATC that we understood that we were given a descend clearance but ATC said; 'NOPE.'


Second narrative: I was the PM; and we were finishing a turn ZZZ-ZZZ1-ZZZ. I hadn't flown in April; flew one turn in May; and was on my third turn in June. This turn was by far the longest; scheduled for X+10 block hours. Although I felt fully prepared for the duty day; I did feel slightly rusty; due to the lack of recent flying. The FO; as PF; also had not flown much recently. We had talked about being tired; the two long legs; and anticipating the end of the day.The radios were busy with the departures and arrivals at ZZZ and ZZZ2; and other area airports. On the ZZZZZ Arrival; shortly after passing ZZZZZ at 6;000 feet; I checked in with  Approach; instructed us to proceed to ZZZZZ1; for the XR Localizer. I read back that clearance. I think I expected a descent out of 6;000 feet; and the phrase 'X-Right Localizer' became; in my mind; 'cleared to 4;000 feet and intercept the localizer.' When I turned to the FO; I told him the clearance was to proceed to ZZZZZ1; descend to 4;000 and intercept the localizer.The FO set 4;000 feet on the MCP; selected VERT SPD; and dialed in about 1;000 FPM descent rate. He then put ZZZZZ1 on the top of the RTE Page 1; and I confirmed both the altitude change and mode; and the route adjustment. The FO executed in the MCDU; and we began a descent and turn to ZZZZZ1. The FO stated he would use LNAV to intercept and then select VOR/LOC afterwards.Passing about 5;200 feet; Approach told us he had not cleared us to descend; and to level at 5;000 feet. I acknowledged that; and replied I thought he had cleared us to descend. Approach replied with a curt 'Nope.' The FO arrested the descent at 5;000 feet; while we attempted to recognize the error. I was convinced we had been cleared to descend; that I had read back a descent clearance; and Approach had not corrected an improper read back. At that point I was confused.About two minutes later; Approach advised us to look for westbound traffic at 4;500 feet to the north. We did not acquire a visual on said traffic. Subsequently we were cleared for the ILS XR; at or above 4;000 feet and 250 knots. I do not recall being directed to contact ZZZ Tower. Five minutes after being cleared for the ILS XR; I called Approach Control in the blind of my intention to contact ZZZ Tower. We landed without further problems.Having previously discussed our tiredness; I should have been on guard for the increased potential for errors. I should have verified the clearance with the FO; PF; before we changed our flight path; instead of telling him what I believed the clearance to be.


Synopsis: Air carrier flight crew reported an altitude deviation during arrival. Captain stated rustiness due to lack of flying was a contributing factor.