Whilst it is true that ATC are primarily responsible for correct vertical separation of en-route IFR traffic, TCAS was introduced as an essential safety complement for those rare occasions when ATC errors occur.
But TCAS is only of any use if the crews know how to use it correctly!!
TCAS RA directions always override ATC instructions!
On 1 Jul 2002, the Bashkirian Airlines Tu 154 and DHL Boeing 757 were on a collision course at FL360. This is what happened in the 50 seconds following the first TCAS Traffic Advisories:
21:34:42 Both aircraft receive TAs
21:34:49 ATC instructs Tu 154 to descend to FL350
21:34:56 Boeing 757 issued “Descend, descend” RA. Crew responds correctly
21:34:56 Tu 154 issued “Climb, climb” RA. Crew continues to descend
21:35:03 ATC instructs Tu 154 to increase descent rate
21:35:10 Boeing 757 issued “Increase descent” RA. Crew responds correctly
21:35:19 Boeing 757 advises ATC they are responding to TCAS RA
21:35:24 Tu 154 issued “Increase climb” RA. Crew continues to descend
21:35:32 Both aircraft collide overhead Überlingen