Which census?
1841, of course, doesn't mention place of birth.
1871 (usually) is indexed only on county, rather than town.
Places of birth can vary - a hamlet that is known to local people, will often be changed to the nearest town if the people moved away from home.
For example, when I am not in the UK, I tell people I am from Bristol, even though I am really from Portishead, Somerset!

I often search on forename and place of birth and year of birth, when I can't find someone.
Only last night I did that searching for a family Raven, and found them as Rover!