  For almost three decades in the twentieth century, the physics community
believed that John von Neumann had proved the impossibility of completing
quantum mechanics by a local realist, hidden-variables theory. Although Grete
Hermann had raised strong objections to von Neumann's proof, she was largely
ignored. This situation lasted, until John Bell rediscovered that von Neumann's
proof was flawed: a \emph{sufficient} condition for local realism had been
taken as a \emph{necessary} one. Bell subsequently established various
constraints on hidden-variables theories, in the form of inequalities that can
be submitted to experimental test. All performed tests to date have opened some
loopholes. The quest to close them motivated great technical achievements and
ongoing efforts to improve what has already been reached. There is, however, a
rather ironic twist concerning Bell inequalities. On deriving them, Bell also
took a sufficient condition for local-realism, as if it were a necessary one.
As a consequence, even completely loophole-free Bell inequality violations
would not disprove local realism. We argue that Bell inequalities cannot follow
from local-realism alone. The proof is given by constructing three
local-realist models that entail Bell inequality violations.
