Need a better explanation and more detail but it sounds as though the driveway straddles the property line between the two properties, in which case there should be an easement in both deeds granting an easement/right of way to both property owners for use of the common driveway.
I'm not completely clear on the "adverse possession" issue other than generally it takes longer in most states to establish(20 years) and it would not necessarily extinguish any rights granted by an easement.
Please give more details and a clearer explanation if you want a better answer.

