  We consider the question whether a wormhole can be converted into a
non-extremal quasi-black black hole by continuous change of parameters. In
other words, we ask whether "black" wormholes can exist as end points of
families of static wormhole geometries. The answer is negative since the
corresponding limit is shown to be singular. Similar conclusions are valid also
for other types of black hole mimickers such as gravastars and quasi-black
holes without wormhole behavior. Our treatment is model-independent and applies
to any static geometries without requirement of special symmetries. We also
find an asymptotic expression for the Kretschmann scalar for wormholes on the
threshold of horizon formation that can be used as an the bound on proximity of
the configuration to the would-be horizon. The derived bound is very weak for
astrophysical black holes but becomes relevant for microscopic ones. We point
out complementarity between ability of wormholes to mimic black holes and their
ability to be traversable "in practice".
