Impossible to give advice as you haven't described the situation well enough. How did you collect samples used to estimate the descriptive statistics? How representative are your samples? Over what changing conditions (what is the inference space)? Have you investigated the measurement systems? If your measurement system has poor discrimination, then you can under estimate the variation. Do you have an evidence the process is stable?
A disclaimer, I am not a fan of capability indices. There are multiple statistics estimated via sampling, therefore there are confidence intervals around each statistic (and, of course, you are reporting one number...there must at least be an interval) and then you have specifications that are independently derived and potentially pulled from thin air...
"All models are wrong, some are useful" G.E.P. Box