In this paper we obtain essentially best possible stability estimates for a class of inverse problems associated to elliptic boundary value problems in which the role of the unknown is played by an inaccessible part of the boundary and the role of the data is played by overdetermined boundary data for the elliptic equation assigned on the remaining, accessible, part of the boundary. We treat the case of arbitrary space dimension n greater than or equal to 2. Such problems arise in applied contexts of nondestructive testing of materials for either electric or thermal conductors, and are known to be ill-posed.