Good TCAS systems based on TXPDR returns are wonderful in helping to deconflict traffic. Like all systems, they have their limitations and the main limitation is generally the pilot. The second, of course, is that it relies on the other guy using his system correctly as well, and having his transponder on in at least Mode C.
My first reaction on reading your post was that it illustrates the danger of relying on your on-board systems too much and not enough on airmanship. Your TIS-B system is good, but relies on several links - your GPS, the GPS satellite system, the ground station, etc. etc. etc.
Note also the limitations
here.
They include:-
No transponder = no TIS-B target. Always look outside.
...
Pilots flying in visual meteorological conditions (VMC) are reminded that visual contact remains the only means of self separation.
...
All pilots/operators are reminded that the airborne equipment that displays other ADS-B equipped aircraft and transponder equipped aircraft via TIS-B is only for pilot situational awareness. This equipment is not approved as a collision avoidance tool.
The Mark I eyeball remains your best friend and collision avoidance tool.