When subjected to strong electric fields, raindrops in thunderclouds, pendant drops in electrospray mass spectrometry, and planar films form conical tips and emit thin jets from their tips. Theoretical analysis of the temporal development of such electrohydrodynamic (EHD) tip-streaming or cone-jetting phenomena has heretofore been elusive given the large disparity in length scales between the macroscopic drops/films and the microscopic jets. Here, simulation and experiment are used to investigate EHD tip-streaming from a liquid film of finite conductivity. In the simulations, the full Taylor-Melcher leaky-dielectric model, which accounts for charge relaxation, is solved to probe the mechanisms of cone formation, jet emission, and breakup of the jet into small drops. Simulations show that tip-streaming does not occur if the liquid is perfectly conducting or perfectly insulating. A scaling law for sizes of micro-(nano-)scale drops produced from the breakup of the thin jets is also developed.