B737-700
Flight Crew Last 90 Days 204; Flight Crew Type 204
Situational Awareness; Confusion; Communication Breakdown
ATC Issue All Types; Conflict Airborne Conflict; Deviation / Discrepancy - Procedural Published Material / Policy
Air Traffic Control Issued New Clearance; Flight Crew Took Evasive Action; Flight Crew Requested ATC Assistance / Clarification
Procedure; Human Factors
Eastbound at FL370; Denver Center issued us an unsolicited clearance to FL380. We accepted clearance but the F/O noticed a TCAS +1 target approximately 20 miles ahead.  Captain questioned clearance and was told to climb to FL380.  F/O stopped climb and returned to FL370. Visual contact was established with traffic and ATC was again queried about traffic.  Clearance was again issued to FL380 and ATC insisted that there was no traffic for us.  By now traffic passed directly overhead.  We informed ATC that not only did we see traffic; but tracked target on TCAS.  ATC insisted that there was no traffic and no conflict. ATC then queried other airplanes on frequency and asked if they saw traffic at FL380.  All said negative.   As we were issued a frequency change; things were still a bit tense as we were trying to figure out what just happened.  ATC said they figured things out.  He was issuing us clearances for Company Flight Number and thought we were them!  This mistaken identity persisted through at least a dozen exchanges between both Pilots on the frequency. This was an example of similar call signs fooling the Controller for an extended period of time.  Perhaps Marketing can pay attention to varying their flight numbers whilst building the schedule.
B737 First Officer reports being issued us an unsolicited clearance to FL380 while at FL370. The clearance is accepted but a TCAS target at FL380 is detected 20 NM ahead and ATC is queried. ATC indicates there is no traffic but The First Officer returns to FL370 and watches the traffic pass overhead. A callsign mix up by ATC was the issue.