Ground PRIMARY radar performance depends on age and condition of ground equipment and range of A/C and A/C radar reflectivity.
Ground SECONDARY radar performance depends on age and condition of ground equipment and range and strength of A/C transponder transmission.
Not all radar heads have both primary and/or secondary radar and some primary displays are fed secondary info by landline from a differently located secondary radar site. To my knowledge this happens alot at rural/regional French airports.
So it is possible for an aircraft to be outside the performance criteria for the primary radar and if ground has no secondary it would be "invisible" for all modes of transponders setting, if switched on. Or that ATC centre might be receiving a secondary feed from a site not "transponding" with the target A/C.
With 2 airborne A/C in relative close proximity they have an ideal "line of sight" transponder environment. If the primary "invisible" contact has a weak transponder strength it may not show on the ground secondary receiver approaching the limit of the ground performance envelope.
And then of course there's the military. Rumour (and I stress rumour) has it that they can vary the strength of their transponder so as not to be visible to ground radars for training and excersice but still be a good "TCAS" rule observer.
One thing that I dont know, and perhaps someone out there can supply, is is military IFF compatible with civil TCAS? Because IFF is not the same system, to my knowlegde, it's more complex and more secure.