Well, the directional qualities of the antennae are (if memory serves) only specified for +/- 15 degrees in azimuth (although many will function well beyond that, 90 degrees might be pushing it). As you don't get 'NO BEARING' proximate traffic (only for TAs and RAs) I guess that might explain it.
TCAS will display an aircraft with a transponder turned on
It won't always. Another cause for target loss is interference limiting mode, but I wouldn't have though this was the case (kind of short range to be filtered).
If the lower antenna was blocked shouldn't matter, TCAS should contact via the upper antenna, in fact the upper antenna is the default, to avoid indirect transmission paths from the ground where possible.
As far as I am aware, every modern TCAS installation includes upper and lower antenna for transponders and the TCAS itself. There was problematic function of TCAS in its early days due to restricted line of sight from single antenna.
Having said that, on some installations the lower TCAS antenna is not directional at all. Since the upper antenna is the one that usually does all the work, it has to be specified to be directional, a few quid can be saved on the lower antenna. Maybe thats the case on your aircraft.
Note that the other aircraft would still be tracked, and a no bearing TA and/or RA issued accordingly.
Finally of course, maybe your lower antenna actually was broke. Again, since the upper is the main, I would imagine that a u/s or weak lower antenna could be carried for quite a while before anyone noticed (supposition here).
pb