TCAS comes in many different variations. I have flown installations that show only traffic in a range of 2,5NM around you and others that show traffic up to 70NM with level ranges between 2000ft and 9000ft (the latter one could be toggled as Rainboe said), traffic displays included in the usual NAV-Display or as after-market installation in the VSI (737 classics).
Usually if the traffic is of any concern to us and within 2000ft of our actual level chances are high we can see it on TCAS when pointed out by ATC, however that is not a foolproof, approved or recommended operation of the TCAS display. When we do get a traffic information we have to check outside the window using our old but reliable eyeballs. Relative bearing, range and altitude is what helps us to identify the target.
By the way, at the moment in europe at least ADS-B in is not really used by airliners, only ADS-B out via Mode S. So we cannot see any flight IDs or additional info, just the position and its threat to us.