Electron clouds in the beam pipe of high-energy proton or positron storage rings can give rise to significant incoherent emittance growth, at densities far below the coherent-instability threshold. We identify two responsible mechanisms: namely, (1) a beam particle periodically crosses a resonance and (2) a beam particle periodically crosses a region of the bunch where its motion is linearly unstable. Formation of halo or beam-core blow up, respectively, are the result. Key ingredients for both processes are synchrotron motion and electron-induced tune shift. The mechanisms considered provide a possible explanation for reduced beam lifetime and emittance growth observed at several operating accelerators. Similar phenomena are likely to occur in other two-stream systems.