As stated by previous posters, the exact distance you get a Resolution Advisory (the command to climb or descend to avoid the intruder) is dependent on the software load. The older 6.04 gave you an alert whenever the flight path came within about 1 NM (this varies a lot with closure rates & relative altitudes) but the later Change 7 is designed NOT to give a false alert when aircraft are in RVSM airspace (as close as 1000ft apart in altitude). Under test conditions (trying to get alerts on the TCAS display on a HUD), it was found you had to be pretty much on a collision course before an RA occurred if Change 7 was loaded.
TCAS uses "whisper/shout" to distinguish between potential intruders. Targets which transmit returns when a high-power interrogation is sent out are ignored for the purposes of conflict management if they do not reply to the low power "whisper" as they are assumed to be out of conflict range. Only those intruders that are tracked and enter the warning zone - the "Tau" - will cause a "TRAFFIC" alert and subsequently give rise to an RA.