On the Gaussian sum of Gaussian variates, the non-Gaussian sum of Gaussian variates, and the Gaussian sum of non-Gaussian variates

The random variable generated by adding two Gaussian variables may or may not have a Gaussian distribution. Also, the random variable generated by adding two non-Gaussian variables may or may not have a non-Gaussian distribution. Of several examples given, one illustrates how the sum may be Gaussian while the individual variables are not.