How credible were the recorded birth dates and locations in Naturalization paperwork?

Question

Daniel’s surname was recorded as Burns by the Gloucester Massachusetts clerk completing the naturalization paperwork in 1888, he signed the document Daniel Byrne. The date of birth was recorded as Shrove Tuesday 1856 in Country Cork rather than 5 February 1856. The two men sworn as witnesses on the naturalization paperwork were both from Dungarvan, County Waterford. 
Daniel married Bridget Hogan at Dungarvan in 1876, the marriage record listed Buttery as the residence, their son John William Byrne was born at Dungarvan in 1880, Buttery was the listed residence. There was a Daniel Byrne born to John Byrne at Dungarvan in 1844, again Buttery was the recorded residence. 
John William Byrne had a son named William Edward Burns at Charlestown Massachusetts in 1911, it is believed he was named after a relative from Dungarvan that was a Policeman in Gloucester. 

Answer 1

As Shrove Tuesday can be anywhere from 3 February to 9 March.

Comments

Tuesday preceding Ash Wednesday in 1856 fell on the 5th. It seems a strange way for the annotation of a birth date. I have not seen this format before. Thank you.