A hearback readback problem between ATC and a B757-200 flight crew regarding the flight's cleared altitude resulted in a possible loss of separation. The failure of the Controller to correct the flight crew's incorrect readback was cited as a factor.

Narrative

During climb we were issued several altitudes. One of the issued altitudes was a climb to FL350 which we heard and read back. I remember this clearance since I thought the Controller might have said 390; but since the First Officer (Pilot Not Flying) read back FL350 and the Controller didn't correct it; I had to assume it was correct. On TCAS after we leveled at FL350 I could see traffic ahead at our altitude. The distance was about 10 miles; though I had to range down from my large range to try to estimate what the Controller had in mind; since we were .80 mach and overtaking the traffic. After we were level for a couple of minutes the Controller said; 'say rate of climb.' Since we were level; the First Officer responded we were level at FL350; our cleared altitude. Right away; we were the 'bad' guys in the Controller's mind. 'I cleared you to FL390. Climb and maintain FL390 and give me a good rate through FL370.'This is another of a long line of Controllers not listening to read backs. If I wrote a safety report for every time this happened I would be writing one every day. I happen to have previously been a Full Performance Air Traffic Controller. One of the tasks that my instructor at the time stressed was to 'listen' to the readbacks. After all; the primary mission of a controller is to 'insure' separation. He also said; that as a Center Controller you will have to say every thing twice. Yes; with cockpit noise; distractions; different call signs every leg; we miss transmissions; and sometimes hear them wrong. However; the fact remains; that it is the 'requirement' of the Controller to 'insure' separation. And; they cannot do this unless they 'listen'. That is why we read back; that which we thought we heard.