We performed a search for short gravitational wave bursts using about 3 years of data of the resonant bar detectors Nautilus and Explorer. Two types of analysis were performed: a search for coincidences with a low background of accidentals (0.1 over the entire period), and the calculation of upper limits on the rate of gravitational wave bursts. Here we give a detailed account of the methodology and we report the results: a null search for coincident events and an upper limit that improves over all previous limits from resonant antennas, and is competitive, in the range , with limits from interferometric detectors. Some new methodological features are introduced that have proven successful in the upper limits evaluation.