We present the results of a search for transient radio bursts of between 0.125 and 32 ms duration in two archival pulsar surveys of intermediate Galactic latitudes with the Parkes multibeam receiver. 14 new neutron stars have been discovered, seven of which belong to the recently identified 'rotating radio transients' (RRATs) class. Here, we describe our search methodology, and discuss the new detections in terms of how the RRAT population relates to the general population of pulsars. The new detections indicate (1) that the Galactic z-distribution of RRATs in the surveys closely resembles the distribution of pulsars, with objects up to 0.86 kpc from the Galactic plane; (2) where measurable, the RRAT pulse widths are similar to that of individual pulses from pulsars of similar period, implying a similar beaming fraction; and (3) our new detections span a variety of nulling fractions, and thus we postulate that the RRATs may simply be nulling pulsars that are only 'on' for less than a pulse period. Finally, the newly discovered object PSR J0941−39 may represent a link between pulsars and RRATs. This bizarre object was discovered as an RRAT, but in follow-up observations often appeared as a bright (∼10 mJy) pulsar with a low nulling fraction. It is obvious therefore that a neutron star can oscillate between being an RRAT and a pulsar. Crucially, the sites of the RRAT pulses are coincident with the pulsar's emission, implying that the two emission mechanisms are linked, and that RRATs are not just pulsars observed from different orientations.