  We present a general methodology to construct triplewise independent
sequences of random variables having a common but arbitrary marginal
distribution $F$ (satisfying very mild conditions). For two specific sequences,
we obtain in closed form the asymptotic distribution of the sample mean. It is
non-Gaussian (and depends on the specific choice of $F$). This allows us to
illustrate the extent of the 'failure' of the classical central limit theorem
(CLT) under triplewise independence. Our methodology is simple and can also be
used to create, for any integer $K$, new $K$-tuplewise independent sequences
that are not mutually independent. For $K \geq 4$, it appears that the
sequences created using our methodology do verify a CLT, and we explain
heuristically why this is the case.
