Suppose $X_i, \; i=1,2,3...,n$ are each Gaussian, then it is not in general true that the set is jointly Gaussian (a multivariate Gaussian).

Does a similar statement hold if the variates are *pairwise* Gaussian? i.e. if we have that $X_i, X_j$ are a bivariate Gaussian for all $i, j$, then it is not in general true that the set is jointly Gaussian. (And I suppose any dimensional variant of this? k-wise Gaussian => n-wise Gaussian)

If the statement holds, what is a simple low-dimensional example where you have a pairwise Gaussian system where this does not hold? Maybe this is in Counterexamples in Probability ... but I don't have this available right now.