As assembly is provided for a gas turbine engine. This assembly includes a component and a brush seal. The component includes a first surface and a second surface. The brush seal includes an annular bundle of bristles which extends longitudinally along an axis to a bundle tip. The bundle of bristles axially contacts the first surface at the bundle tip. The bundle of bristles proximate the bundle tip axially overlaps the second surface, and the bundle of bristles is operable to radially contact the second surface.