For $n>2$ variables, one cannot arbitrarily choose the correlations $\rho_{ij}$ because the resultant correlations must obey the law of cosines. Equivalently, the covariance matrix between them must be positive definite.
Suppose I have $n>2$ variables, each of which has a correlation coefficient with respect to each other variable $\rho_{ij}=r$ for some fixed $r$ for all of the variables. That is, the correlation matrix is defined by ones on the diagnonal and r on every off-diagonal entry. What is the maximum and minimum value of $r$?
I have not been able to find an analytic solution to this. There are general recursive formulae for constructing correlations in $n$ dimensions so that they are all consistent, but I have not seen an answer to this problem despite looking.
(This is not homework!)