Narrative:

Original clearance 310 degree heading and 1600 ft. Our aircraft was cleared for takeoff on runway 36R at tpa with an amended heading of 'runway heading.' immediately after takeoff, the tower pointed out traffic near us and our altitude and informed us that he would turn us as soon as he got some altitude from us. The first officer called the traffic 'in sight' near us, then lost contact. The TCASII indicated the aircraft was closing in and the alarm went off. I started climbing to avoid the traffic. I stopped at 2200 ft MSL then returned to 2000 ft MSL and informed departure (as we were switched over by tower) of our altitude. The first officer told departure we were at 2000 ft and that tower had told us to climb to avoid traffic. Departure control hesitated, but did not question further. Our assigned altitude had been 1600 ft MSL, but due to the encroaching traffic and the inference by tower that he needed a climb, I breached the altitude. Rather than returning to 1600 ft, due to the confusion, I thought it better just to inform departure of our altitude. We were promptly cleared to a higher altitude and no further discussion took place. After the first officer's initial contact with the other traffic and subsequent loss of it, it was not seen again.

Related reports:

Google
 

Original NASA ASRS Text

Title: AN ACR BE02 FLC RECEIVES A TA FROM THE TWR CTLR AND THEIR TCASII AND PROPERLY REACT.

Narrative: ORIGINAL CLRNC 310 DEG HDG AND 1600 FT. OUR ACFT WAS CLRED FOR TKOF ON RWY 36R AT TPA WITH AN AMENDED HDG OF 'RWY HDG.' IMMEDIATELY AFTER TKOF, THE TWR POINTED OUT TFC NEAR US AND OUR ALT AND INFORMED US THAT HE WOULD TURN US AS SOON AS HE GOT SOME ALT FROM US. THE FO CALLED THE TFC 'IN SIGHT' NEAR US, THEN LOST CONTACT. THE TCASII INDICATED THE ACFT WAS CLOSING IN AND THE ALARM WENT OFF. I STARTED CLBING TO AVOID THE TFC. I STOPPED AT 2200 FT MSL THEN RETURNED TO 2000 FT MSL AND INFORMED DEP (AS WE WERE SWITCHED OVER BY TWR) OF OUR ALT. THE FO TOLD DEP WE WERE AT 2000 FT AND THAT TWR HAD TOLD US TO CLB TO AVOID TFC. DEP CTL HESITATED, BUT DID NOT QUESTION FURTHER. OUR ASSIGNED ALT HAD BEEN 1600 FT MSL, BUT DUE TO THE ENCROACHING TFC AND THE INFERENCE BY TWR THAT HE NEEDED A CLB, I BREACHED THE ALT. RATHER THAN RETURNING TO 1600 FT, DUE TO THE CONFUSION, I THOUGHT IT BETTER JUST TO INFORM DEP OF OUR ALT. WE WERE PROMPTLY CLRED TO A HIGHER ALT AND NO FURTHER DISCUSSION TOOK PLACE. AFTER THE FO'S INITIAL CONTACT WITH THE OTHER TFC AND SUBSEQUENT LOSS OF IT, IT WAS NOT SEEN AGAIN.

Data retrieved from NASA's ASRS site as of July 2007 and automatically converted to unabbreviated mixed upper/lowercase text. This report is for informational purposes with no guarantee of accuracy. See NASA's ASRS site for official report.