I have problems to interpret dependencies in the covariance matrix $Cov(\epsilon)$ of errors in the linear Regression model $Y = X\beta + \epsilon$, i.e. when there are non-zero entries apart from the entries in the diagonal. For example let's assume that $Cov(\epsilon_i,\epsilon_j) > 0$ for some $i$ and $j$. If I understand the concept of errors right, $Cov(\epsilon_i,\epsilon_j) > 0$ implies that the deviations from the expected $y_i$ and $y_j$ value (i.e. the expected value given the regressor values in the $i$th and $j$th row of $X$), are somehow positively correlated.
Unfortunately, I cannot attach any meaning to this beyond this mathematical interpretation (if it is correct at all). As I understand it, $\epsilon_i$ and $\epsilon_j$ belong to two different "populations" that are defined by the respective regressor values in the $i$th and $j$th row. Why should there be an association between the error terms then? I appreciate any help.