We examine excitation (by a short laser pulse) of a hydrogen atom in parallel electric and magnetic fields, from an initial tightly bound state to a state above the classical ionization threshold. We predict that the atom ionizes by emitting a train of electron pulses. This prediction is based on the classical dynamics of electron escape. In particular, the pulse train is due to classical chaos, which occurs for nonvanishing magnetic field. We connect the structure of the pulse train to fractal structure in the escape dynamics, and discuss several issues of experimental interest, with a particular emphasis on understanding the resolution of individual pulses. A brief account of this work appeared previously as a Letter [Phys. Rev. Lett. 92, 073001 (2004)].