  In order to physically enlighten the relationship between {\it
$q$--independence} and {\it scale-invariance}, we introduce three types of
asymptotically scale-invariant probabilistic models with binary random
variables, namely (i) a family, characterized by an index $\nu=1,2,3,...$,
unifying the Leibnitz triangle ($\nu=1$) and the case of independent variables
($\nu\to\infty$); (ii) two slightly different discretizations of
$q$--Gaussians; (iii) a special family, characterized by the parameter $\chi$,
which generalizes the usual case of independent variables (recovered for
$\chi=1/2$). Models (i) and (iii) are in fact strictly scale-invariant. For
models (i), we analytically show that the $N \to\infty$ probability
distribution is a $q$--Gaussian with $q=(\nu -2)/(\nu-1)$. Models (ii) approach
$q$--Gaussians by construction, and we numerically show that they do so with
asymptotic scale-invariance. Models (iii), like two other strictly
scale-invariant models recently discussed by Hilhorst and Schehr (2007),
approach instead limiting distributions which are {\it not} $q$--Gaussians. The
scenario which emerges is that asymptotic (or even strict) scale-invariance is
not sufficient but it might be necessary for having strict (or asymptotic)
$q$--independence, which, in turn, mandates $q$--Gaussian attractors.
