Even the Honeywell K-mart version TCASII unit predicts issues warnings on traffic out to 15-35 seconds and cautions on traffic 20-48 seconds from the collision area (from the manual). This equates to more than 5-8 miles and most sets are better than this.
Switch-selectable limits allow display of traffic from "2700ft above or below to 9900ft ".
Not teaching anyone to suck eggs but if the J* was more than a couple of thousand feet below you, out at "5-8" miles he would appear to be about the same level. Maybe you were just outside his vertical range.