I wouldn't be surprised if birth info on a death cert was wrong, but do you mean we cannot even trust the death date?
The old Swedish church records that are my main sources contain quite a lot of inexactitudes and outright errors - it usually helps to get to know the parish, or at least the whole family, rather than just dive in for the data of a single person. Since deaths were usually recorded by the same person who performed the burial, date errors are rare - but name errors are quite common with the more absentminded vicars. It is not just the descendants of emigrants who get confused by the patronymic system - there are vicars who mix up Anders Eriksson and Erik Andersson as well.