The laws on this vary from state to state, but generally speaking, you do not "cede" your property by not building your fence on the actual property line. What the poster is referring to is a doctrine known as "adverse possession" whereby a person may obtain title to land that they did not own under certain circumstances.
They must occupy/maintain the property OPENLY, ADVERSELY and NOTORIOUSLY for the statutorily mandated period of time (in my state it is 15 years). An adverse possession case will fail if it can be shown that they have not met any one of the statutory requirements. For instance, even if your neighbor mows the other side of the fence (ie., he maintains it) he has not done so "adversely" if you permitted him to do so.
If you build the fence ON the boundary line it then becomes a "boundary fence" and in most states, the fence then becomes the common property of both parties (ie., he can attach his fence to it if he wishes) and both parties are responsible for maintaining the fence. By building your fence 1 foot off of the line (on your side) it is no longer a boundary fence and is instead what is referred to as a fence of convenience and it belongs to you and your neighbor has no rights or responsibilities regarding the fence whatsoever.
I am a licensed professional land surveyor licensed in two states and have a degree in Land Surveying. This is my pro-bono contribution for the day