In a large laboratory plasma a current sheet is generated in the process of magnetic field line reconnection. The stability of the sheet with respect to local current increases is investigated. When the current density in the center of the sheet exceeds a critical value, spontaneous local current disruptions are observed. The current from the center of the sheet moves out to the sides. Magnetic flux variations in regions remote from the current sheet generate an inductive voltage in the current loop that drops off inside the plasma in the form of a potential double layer. This leads to particle acceleration with velocities much larger than those expected from the steady state electric fields in the plasma. The particle beams acquire their energy at the expense of the stored magnetic field energy of the current system. Beam-plasma instabilities are generated that dissipate some of the directed kinetic energy and heat the background plasma. A model for the mechanism of the current disruptions is formulated. The potential structure leads to ion expulsion creating a localized density drop. The associated current drop in an inductive circuit drives the potential structure, thereby providing feedback for the disruptive instability. It saturates at a total current loss upon which the current system recovers, and the process repeats randomly. Similarities and differences to magnetospheric substorm phenomena are pointed out.