We have made observations of the H I in the southern ringed barred spiral galaxies NGC 1433 and NGC 6300 with the Australia Telescope Compact Array (ATCA), the main goal being to test the resonance theory for the origin of these rings. NGC 1433 is the prototypical ringed barred spiral and displays distinct H I counterparts to its nuclear ring, inner ring, outer pseudoring, and plumelike features. The L(4) and L(5) regions at corotation, as well as the bar itself, are relatively devoid of neutral atomic hydrogen. We use the Tully-Fisher relation to argue that the mean inclination of the disk of NGC 1433 is closer to 33 degrees than to 20 degrees, meaning that its outer pseudoring is intrinsically almost circular, while the inner ring is rather more elongated than the average (based on results from the Catalog of Southern Ringed Galaxies). Strong radio continuum emission is localized to the nucleus and the ends of the bar in NGC 1433, and we place an upper limit on the 1.38 GHz flux of the Type II SN 1985P a decade after the explosion, By associating the inner ring of NGC 1433 with the inner second harmonic resonance, and its outer pseudoring with the outer Lindblad resonance, we are able to infer a bar pattern speed for NGC 1433 of 26 +/- 5 km s(-1) kpc(-1). By way of contrast, NGC 6300 possesses a much more extended H I disk than NGC 1433, despite having a similar morphological type. There is a gas ring underlying the inner pseudoring, but it is both broader and slightly larger in diameter than the optical feature. The outer H I envelope has a 20 degrees kinematic warp as well. as a short tail, even though there are no nearby candidates for a recent interaction with NGC 6300. The noncircular motions inferred from optical emission-line spectra do not appear to extend beyond the bar region of NGC 6300. Barely 10% of the 1.38 GHz radio continuum emission in NGC 6300 originates in the type 2 Seyfert nucleus, with the rest coming from a disklike component internal to the ring. By again linking an inner ring feature to the inner second harmonic resonance, we derive a bar pattern speed for NGC 6300 of 27 +/- 8 kM s(-1) kpc(-1), but in this case, neither the outer pseudoring nor the nuclear ring predicted by the resonance ring theory can be identified in NGC 6300. Although it may be the case that the ring in NGC 6300 is not related to a resonance with the bar at all, we postulate instead that NGC 6300 is merely a less well developed example of a resonance ring galaxy than is NGC 1433.