Cpk clears 1.33 but the customer still finds bad parts — where do you look first?

This argument comes up on nearly every capability review I have sat in, and I have never seen it settled the same way twice. So it may as well be the first question on this forum.

The situation, in the abstract: a machined feature reports a Cpk comfortably above the 1.33 the customer asks for. The study looks clean. The gauge has been re-qualified. There is no obvious tool-wear pattern inside a shift. And the customer still pulls parts at incoming inspection.

The usual suspects, roughly in the order people reach for them:

• Gauge correlation — your gauge and their CMM disagree, and neither is wrong on its own terms
• Sample size — a 30 or 50 piece snapshot never sees the long-term spread, so Cpk flatters you where Ppk would not
• Subgrouping — rational subgroups chosen badly make sigma-within look smaller than the process really is
• Drift the study never spanned — shift change, coolant temperature, a re-sharpened tool

What I would like from people who have actually had to close this argument with a customer, not just describe it:

1. Which of these do you check first, and what makes you jump straight to one rather than working down the list?
2. When it did turn out to be gauge correlation, what evidence actually convinced them — short of shipping parts back and forth for weeks?

If you have a case where the answer surprised you, that is the most useful thing you could post here. Numbers welcome.