Flexural vibrations of thin metal wires owing to a high, pulsed electric current have been investigated. The current is sufficiently low to inhibit melting but sufficiently high to induce stresses, leading to the wire fragmentation. The problem is treated numerically on the basis of the theory of three-dimensional linear elasticity. The model has been verified on the well-known exact, eigenmode solution for the flexural vibrations of an infinite wire. The agreement is excellent. Further, the model has been used to study vibrations owing to two sources. The first one is perturbations of wires owing to the Lorentz force leading to a kink-type instability similar to that in plasmas. As the main cause of the wire fragmentation has been previously found to be the thermal expansion of material owing to Joule heating, this problem mainly serves to compare results between the three-dimensional and the one-dimensional, thin-rod models. Comparison of the growth rate of the instability obtained by the two models has shown an excellent agreement. The second source of vibrations is the magnetic fields induced in the external electric circuit. The results show that depending on the shape of the circuit, the induced stresses may exceed 20 MPa for the aluminium wires used in the low-current experiments. Although the external fields are not the main source of the wire fragmentation, these values alone may cause the fracture process at elevated temperatures.