The formation of protein fibrils, and in particular amyloid fibrils, underlies many human diseases. Understanding fibril formation mechanisms is important for understanding disease pathology, but fibril formation kinetics can be complicated, making the relationship between experimental observables and specific mechanisms unclear. Here we examine one often-proposed fibril formation mechanism, nucleated polymerization with off-pathway aggregation. We use the characteristics of this mechanism to derive three tests that can be performed on experimental data to identify it. We also find that this mechanism has an especially striking feature: although increasing protein concentrations generally cause simple nucleated polymerizations to reach completion faster, they cause nucleated polymerizations with off-pathway aggregation to reach completion more slowly when the protein concentration becomes too high.