Eligibility to vote in a municipality is based on the incorporated boundaries of said municipality. Postal addresses are based on a separate (nationwide) system designed for mail delivery, not around municipal boundaries, therefore you can have an address listed as being in a city without actually living in the municipality.
A postal address must include a city, despite not every residential address actually being a part of one. Hence, postal addresses retain their own geographic areas referred to as ‘cities’, that may or may not be actual municipalities, in order to ensure all addresses used for delivering mail have an associated ‘city’. These ‘cities’, even if they share a name with a municipality, are never fully congruous with municipal boundaries, and actually refer to the name of the post office to which mail destined for a particular postal address is associated with.
For example, most people living in the municipality of Swepsonville actually have a Graham, NC address, as the postal version of ‘Graham’ extends well into the Swepsonville municipality, and is served by the Graham Post Office. But if they were to vote in a municipal election, they would receive a Swepsonville ballot, not a Graham one, because they don’t actually live within the municipal boundaries of Graham, they live in Swepsonville, a separate municipality.
Often, people will tell you that the way to know if you actually live in a municipality or not is that if you pay taxes to the city, you live in the municipality, and if you don’t, then you don’t. What people fail to mention when using that explanation is that by ‘taxes’ they mean property taxes. Utilities, trash payments, and (depending on municipal law) certain types of purchases include taxes that are paid to a municipality, but are part of payments for services rendered by the municipality. You may pay Burlington utilities because they are the closest source of water to your residence, but not actually live in the Burlington municipality.
Another issue with the property taxes explanation is that requiring someone to own property to vote in any election is illegal and unconstitutional at a Federal and State level. Paying property taxes to a city is just an indication that you do in fact live within the boundaries of the city, not a requirement for voting in a municipality. It is also possible to live in an apartment or rental property within the boundaries of Burlington, and you can vote if that is the case for you, but in that case you wouldn’t be paying property taxes.
Ultimately residential eligibility for voting in a municipal election all comes down to your permanent residence’s actual geographic location, which is determined through GIS mapping. If a Board of Elections office tells you that you cannot vote in a municipality, that determination is based on the physical location of your residential address.
As always, if you believe there has been a mistake and you do in fact live in a municipality, you are welcome to vote a provisional ballot if this determination will otherwise prevent you from voting at a polling place.