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We analyze the observed properties of nested and single stellar bar systems in disk galaxies. The 112 galaxies in our sample comprise the largest matched Seyfert vs. non-Seyfert sample of nearby galaxies with complete near-infrared or optical imaging sensitive to lengthscales ranging from tens of pc to tens of kpc. We find that a significant fraction of the sample galaxies, 17% +/- 4%, has more than one bar, and that 28% +/- 5% of barred galaxies have nested bars. The bar fractions appear to be stable according to reasonable changes in our adopted bar criteria. For these nested bars, we detect a clear division in length between the large-scale (primary) bars and small-scale (secondary) bars, both in absolute and normalized (to the size of the galaxy) length. We argue that this bimodal distribution can be understood within the framework of the inner Lindblad resonances (ILRs), which are located where the gravitational potential of the innermost galaxy switches effectively from 3D to 2D. While primary bars are found to correlate with the host galaxy sizes, no such correlation is observed for the secondary bars. Moreover, we find that secondary bars differ morphologically from single bars. Overall, our matched Seyfert and non-Seyfert samples show a statistically significant excess of bars among the Seyfert galaxies at practically all lengthscales. We confirm our previous result that Seyfert galaxies always show a preponderance of "thick" bars compared to the bars in non-Seyfert galaxies. Finally, no correlation is observed between the presence of a bar and that of companion galaxies, even relatively bright ones.