Cell adhesion initiates with ligand receptor binding, but the immediate consequences are typically also of interest. This can mean signaling, cell rolling and arrest, engulfment or phagocytosis, or endocytosis (for instance relevant to the uptake of drug delivery particles and viruses.) This paper concerns situations where an adherent object or cells is (partially) engulfed by the cell of interest. Such a process involves substantial rearrangement of the cell membrane and underlying cytoskeletal structure. With a series of bilayer polymer membranes, we demonstrate a coupling between adhesion and membrane mechanics which produces qualitative differences in the resulting ability of a “cell” (polymer vesicle) to engulf a target, including metastable binding states and arrested spreading.