Real Estate Law
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Your neighbors are referring to adverse possession, which is when someone gains ownership to property by occupying property and claiming ownership for a certain period of time. In New York, the time period is 10 consecutive years (New York Real Prop. A&P.L. §501-551).
Thus, if the fence has been there for at least 10 years, the neighbor is likely correct in what he/she is telling you.
Whether this impacts your desire to purchase the house is up to you, but you don't stand to recover the property that has been acquired and enclosed with a fence by your neighbor.
If you do proceed, however, I would recommend that you make sure that your neighbor is paying taxes on that portion of the property - not you.