TCAS has a range of sensitivity bands that are switched in at different altitudes. It all depends what altitude you were at as to whether you would get an RA or not.
Typically,for 2 aircraft flying level below 5000ft, seperation of 300ft or less will generate an RA, anything greater and a traffic only warning will be given. Between 5000ft and 10000ft 350ft is min seperation rising to 600ft at 20,000ft to 42,000ft.
All this assumes level flight of both aircraft, if either starts climbing or descending its a whole new ballgame!