It seems everyone in this thread agrees that the determination for the appropriate sample size for statistical analysis stems from what your desired confidence and reliability levels. At the company I work for we use our confidence and reliability to determine how we evaluate specific product requirements based on whether or not the test we are performing is variable or attribute.
For attribute testing the confidence and reliability levels gets translated into a minimum number of samples you need to test and out of those tested samples how many failures you are allowed to have. For example for a given confidence level “X” and reliability level “Y” the internal procedure would state that you need to test a minimum of 150 sample and are not allowed to have any failures. However, this would not be your only option for your attribute test the procedure would also state higher sample sizes in which you are able to accept more failures. For instance for the same confidence and reliability level (“X” and “Y”) you may have to test 320 sample to be able to accept one failure. It works as a sliding scale, and for any given confidence and reliability level the more samples you test above the minimum will start to allow you to accept more failures.
For variable testing the confidence and reliability levels are translated into an acceptable capability value (CpK or PpK). Based on the mean, standard deviation, and specific product requirement value the capability can be determined. For my company which produces high volume disposable medical devices the minimum samples size (as dictated by our corporate statistics department) is N=30 samples in order to determine normality and capability. You can always test more if you can afford it based on the timeline and cost associated with the testing for your given product but N=30 would be the minimum. Generally from my experience there is really no negative will testing more samples, but testing only 30 can sometimes pose potentially issues even though it is statistical relevant.