The confusion might be that the family lived in one county but the postal town/registration district, etc. was actually in the other county so when asked what county are you from both answers could be given.
I agree aghadowey - also a parish/diocese could influence;
not to mention the GAA!
LinkThere are many towns in Ireland especially on rivers that are divided
The Local Government (Ireland) Act 1898 ensured that all of Drogheda was moved into Louth.
It was previously partly in Co. Louth and Co. Meath
Several areas had the boundaries agreed by this act
Carrick-on-Shannon is in Co. Leitrim but right on the border (the train station is in Roscommon!)
Athlone borders Roscommon/Westmeath
Ballinasloe borders Galway/Roscommon
Bray already mentioned borders Dublin / Wicklow etc