We have examined the possibility that rapid weakly magnetized pulsars might be formed in Type II supernovae such as SN1987A in the Large Magellanic Cloud. Since these supernovae are the expected progenitors of the strongly magnetized galactic disc pulsars, it is then necessary to assume that the magnetic field grows rapidly and saturates at about 10(12) G. Given that the Crab pulsar already has a strong field after 900 yr, there is only a short time in which to grow the field. We show that the fast pulsar does not have time to spin down significantly during the growth period, with the result that we will have for a while a fast strongly magnetized pulsar, with a luminosity 1000 times that of the Crab pulsar. This bright phase should last several hundred years, roughly the interval between supernovae, which raises the question why they have not yet been detected. In any event, the luminosity should increase steadily, by about 2 per cent yr-1 or possibly more in an exponential-increase model, and by as much as 38 per cent yr-1 in a linear-increase model. Why the 'millisecond' pulsars themselves do not experience such an increase of magnetic field is an open question. It is potentially interesting, however, that the time to reach saturation is of the order of a few hundred years in both models. Our overall conclusion is that a rapid weakly magnetized pulsar- in SN1987A appears to be inconsistent with what is known about pulsars and supernova remnants (SNRs), but is not entirely excluded. Moreover, a small but steady continued increase in pulsar magnetization would account for the observations of braking indices less than 3 in all young pulsars where it can be determined.