  We construct explicitly two sequences of triplewise independent random
variables having a common but arbitrary marginal distribution $F$ (satisfying
very mild conditions) for which a Central Limit Theorem (CLT) does not hold. We
obtain, in closed form, the asymptotic distributions of the sample means of
those sequences, which are seen to depend on the specific choice of $F$. This
allows us to illustrate the extent of the `failure' of the classical CLT under
triplewise independence. Our methodology is simple and can also be used to
create, for any integer $K$, new $K$-tuplewise independent but dependent
sequences (which are useful to assess the ability of independence tests to
detect complex dependence). For $K \geq 4$, it appears that the sequences thus
created do verify a CLT, and we explain heuristically why this is the case.
