The 1900 U.S. Census asked for both age and month + year of birth. Although the information on month and year of birth is often valuable for genealogists, I have the impression that this feature of the census also introduced quite a few errors because the people who created the records were arithmetically challenged.
Sometimes the date and age are incongruent. For example, a child might be reported as born in November 1889 and age 11, when they were only age 10 as of the date of the census. Was this a miscalculation by the census taker after the family told the census taker that the child was 11 and was born in November, or did the census taker miscalculate age after the family said the child was born in November 1889, or did the family report the age and birth date inconsistently?
I have seen a number of instances where either age or date of birth was later changed on the census sheet to correct an incongruent pair of values (such as changing the age from 11 to 10 when the birth date is recorded as November 1889), but it is not clear what basis the census workers had for deciding which of the two values was wrong.