No, correlation coefficients are not "additive." For example, an average of correlation coefficients in a number of samples does not represent an "average correlation" in all those samples. Because the value of the correlation coefficient is not a linear function of the magnitude of the relation between the variables, correlation coefficients cannot simply be averaged. In cases when you need to average correlations, they first have to be converted into additive measures. For example, before averaging, you can square them to obtain coefficients of determination which are additive (as explained in How to interpret the value of correlations), or convert them into so-called Fisher z values, which are also additive (an r-to-Z transformation procedure is available in the Probability calculator).