  We introduce a general condition sufficient for the validity of the original
Bell inequality (1964) in a local hidden variable (LHV) frame. This condition
can be checked experimentally and incorporates only as a particular case the
assumption on perfect correlations or anticorrelations usually argued for this
inequality in the literature. Specifying this general condition for a quantum
bipartite case, we introduce the whole class of bipartite quantum states,
separable and nonseparable, that (i) admit an LHV description under any
bipartite measurements with two settings per site; (ii) do not necessarily
exhibit perfect correlations and may even have a negative correlation function
if the same quantum observable is measured at both sites but (iii) satisfy the
"perfect correlation" version of the original Bell inequality for any three
bounded quantum observables A, A'=B, B' at sites "A" and "B", respectively.
Analysing the validity of this general LHV condition under classical and
quantum correlation experiments with the same physical context, we stress that,
unlike the Clauser-Horne-Shimony-Holt (CHSH) inequality, the original Bell
inequality distinguishes between classicality and quantum separability.
