What I do not quite understand is mainly related to the fact that for a process to be under control the output is compared with the long-term plus and minus three sigma limits, and, observing an out of control event (beyond these limits) will then call for investigation, possibly stopping the process, trouble-shooting, adjusting the process, etc. (See also: Six Sigma vs. Control Charts based on 3 Sigma Limits )
However, such an event (beyond plus and minus three sigma) can happen 3 out of 1000 times. This is a fairly high rate which I think can present itself as an occasional observation for and in many high volume production processes (including service companies once their output opportunities are considered); right? In other words, I think the three sigma limits are too tight to allow such natural variability, in particular, as will be bound to happen in high volume processes. This is particularly very critical as occurrence of any point beyond the limits are considered to be indicative of out of control/ special cause/shift of the process mean scenarios which, again, will call for strong reaction up to and including adjustment of the process when it may not be needed at all!
However, to add to that, let's say now, if a process is running at Six Sigma level (being under control, with a Cp = Cpk = 2.0), a 2ppb out of specification occurrence is considered to be expected and this process is considered to be running fantastically. Of course, for this process, I guess all the parts beyond three sigma and below Six Sigma are also definitely accepted without any problems anyway process is not considered to have any issues or no investigations/adjustments are called for. This, to me, is contrary with the first principle/test/requirement for a process to be under control (as outlined above, any occurrence beyond three sigma means problem!). Of course, this can further create a fallacy working with Six Sigma (as I have seen it here and there) that as long as the process is meeting the DPMO requirement (2ppb or 3ppm depending if the process is centered), it is running great without even considering the distribution of the output within plus and minus Six Sigma range!
There are a couple issues to consider:
With regard to the assertion that people incorrectly estimate DPMO defect rates and Sigma Level without considering the distribution of the output within plus and minus Six Sigma range, then I would say they are not doing the analysis correctly. There is plenty of that, no doubt.
Learn more about the Lean Six Sigma principles and tools for process excellence in Six Sigma Demystified (2011, McGraw-Hill) by Paul Keller, in his online Lean Six Sigma DMAIC short course ($249), or his online Green Belt certification course ($499).