  Identifying defective items in larger sets is a main problem with many
applications in real life situations. We consider the problem of localizing
defective nodes in networks through an approach based on boolean network
tomography (BNT), which is grounded on inferring informations from the boolean
outcomes of end-to-end measurements paths. {\em Identifiability} conditions on
the set of paths which guarantee discovering or counting unambiguously the
defective nodes are of course very relevant. We investigate old and introduce
new identifiability conditions contributing this problem both from a
theoretical and applied perspective. (1) What is the precise tradeoff between
number of nodes and number of paths such that at most $k$ nodes can be
identified unambiguously ? The answer is known only for $k=1$ and we answer the
question for any $k$, setting a problem implicitly left open in previous works.
(2) We study upper and lower bounds on the number of unambiguously identifiable
nodes, introducing new identifiability conditions which strictly imply and are
strictly implied by unambiguous identifiability; (3) We use these new
conditions on one side to design algorithmic heuristics to count defective
nodes in a fine-grained way, on the other side to prove the first complexity
hardness results on the problem of identifying defective nodes in networks via
BNT. (4) We introduce a random model where we study lower bounds on the number
of unambiguously identifiable defective nodes and we use this model to estimate
that number on real networks by a maximum likelihood estimate approach
