I'm not an expert, but i don't think that's the case. i looked on the school district's website, but couldn't find any policies on residency and enrollment. I know when I registered my kids in bedford county schools, i had to provide two different proofs of address.
it is true that all Bedford County schools are open to any student who is a resident of the county (meaning, my child could be zoned for Cascade, but can choose to attend either Shelbyville Central or Community High School, as long as I am willing to provide transportation). the zoning really only applies to the bus routes. but i don't think that "open school" policy applies to non-Bedford county residents. The exception, i would think, would be if a child's parent worked for the school district, for example, some people in nearby Coffee county are teachers in Bedford county schools.
Sorry I can't offer a more definitive answer...this is just my opinion based on what I've personally experienced here.