Submicrometric magnetic amorphous wires are good candidates for future development of miniaturized sensors and magnetic logic applications. Here we report the results of an in-depth investigation of magnetization switching in rapidly solidified nearly zero magnetostrictive (Co0.94Fe0.06)72.5Si12.5B15 amorphous samples with diameters of the actual magnetic wires between 300 and 450 nm. All samples were found to be magnetically bistable, displaying characteristic rectangular hysteresis loops. This shows that magnetization reversal occurs through the depinning and subsequent propagation of a magnetic domain wall, whose velocity depends on the applied field and on the sample dimensions. The results of this study reveal stochastic nonlinear dependencies of both the magnetic switching field and the domain wall velocity on the sample diameter. The analysis of the potential causes, which include nonlinear residual stresses, fluctuations in wire dimensions (metal and glass), and competing magnetic anisotropies of different origins, show that a combination of all three factors could lead to the observed stochastic behavior. Calculated values of the switching field, which consider only changes in the wire dimensions, indicate that such influence alone cannot account for the strong nonlinearities. The results are important for the applications of such ultrathin cylindrical magnetic amorphous wires.