It's the number of rejects in a 6σ process, assuming a 1.5σ mean shift (1.5σ is an arbitrary assumption for a shift in a long-term process).
You can get a reasonable estimate through random sampling -- which is the basis for most of inferential statistics -- so it's not that flexible. If an adequately powered study was conducted (with a large enough sample size and sufficiently randomized sample), I don't see why the results can't be trusted [2]. A mistake is not that hard to classify (in logistics there are failure reason codes -- wrong address, wrong order, missed delivery, etc. and typically certain categories dominate -- the others are noise).
That said, 3.4 DPMO is kind of a magic number (sort of like 95% confidence intervals -- why 95%? Because Pearson or someone decided it and it stuck).
[1] https://www.qualitygurus.com/why-is-6-sigma-equal-to-3-4-dpm...
[2] Side: Systematic misreporting might be an issue of course, and there's not much that can be done there except design a system of incentives and penalties around it. Customer complaints and lost clients are often visible in such systems.
Case 1 ------ 10 years x 280 deliveries per year x 2 deliveries per day (pick up + drop) = 5600 deliveries.
Number of mistakes = 0
That puts the error rate at < 1/5600 (or approximately 0.0002).
Case 2 ------ 4 years x 280 deliveries per year x 2 deliveries per day (pick up + drop) = 2240
Number of mistakes = 0
That puts the error rate at < 1/2240 (or approximately 0.0004).
Both cases together -------------------
Error rate < 0.00012