I have 13 point predictions from 13 independent linear regressions, each prediction with a 95% confidence interval. I want to sum the 13 predictions and calculate the 95%CI for the summed value. How, or should I, combine the 13 CIs to get the CI for the summed value?
Do you have predictions and prediction intervals (i.e., for future observables), or parameter estimates and confidence intervals (i.e., for unobservable model parameters)? No, it won't make a difference for the answer, but if you have your nomenclature correct, it will be easier for you to find help.
Either way, you have a sum of 13 independent $t$ distributed random variables. Unless you have some specific information on your 13 variables, like common variances, the sum does not have a closed form solution.
You can either simulate many realizations and look at the empirical distribution of the sums, or (if you have sufficiently high degrees of freedom) approximate your $t$ distributions by independent normals and hope for the best. The sum of independent normals is normal, with mean equal to the sum of the component means (same for variances).