A film that is both elastic and water-sensitive (e.g., water-soluble, water-dispersible, etc.) in that it loses its integrity over time in the presence of water is provided. To achieve these dual attributes, the film contains an olefinic elastomer and a water-soluble polymer. Although these polymers are normally chemically incompatible due to their different polarities, the present inventors have discovered that phase separation may be minimized by selectively controlling certain aspects of the film, such as the nature of the polyolefin, water-soluble polymer, and other film components, the relative amount of the film components, and so forth. For example, certain water-soluble polymers may be selected that have a low molecular weight and viscosity to enhance their melt compatibility with nonpolar polyolefins. This, in turn, may result in a film that is generally free of separate phases, which would otherwise limit the ability of the water-soluble polymer to contact water and disperse.