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The excess noise levels in HTS grain boundary Josephson junctions, when expressed as critical current fluctuations |/spl delta/I/sub c/|, vary linearly with I/sub c/ as the temperature changes. By contrast, when the critical current is suppressed by an external magnetic field, the measured fluctuations |/spl delta/I/sub c/| remain nearly constant while L itself undergoes order-of-magnitude changes. This effect, which does not seem to have been widely recognized, is examined with respect to the spatial modulation of the critical current density within the junction. Calculations based on this model are in good agreement with our experimental measurements. In related studies, we measure the excess noise levels between Shapiro steps in a microwave-irradiated junction, and find them to agree with a modified version of the Likharev-Semenov equation for thermal noise.