We introduce a scale invariant gradient norm (SIGN) capable of identifying, and distinguishing between, algorithmic stalling and convergence at a local or global minimum. Our model fitting algorithm accurately estimates all lineshape parameters with superior precision and accuracy compared to the CLS method. We show how to infer when a model has too many, or too few, Kubo components for a data set based on the behavior of the SIGN. Interestingly, we find no evidence of local minima when fitting to a multi-Kubo lineshape of a three-level system. Though analysis of simulated spectra suggests the CLS method is reliable in retrieving Kubo time constants, we have shown an experimental example in which the CLS time constants differ by 60% between independent measurements of the same system. In contrast, Kubo time constants obtained by model fitting only differ by 10%, which suggests model fitting is a far more reliable and consistent means of measuring spectral diffusion over the CLS method. Furthermore, we revealed a fundamental oversight in propagation of error with the CLS method, which led us to show that error bars for Kubo amplitudes and homogeneous dephasing obtained by fitting the linear absorbance spectrum are unreliable. In contrast, model fitting yields reliable error bars over a wide range of scenarios with upwards of 50× better precision than the CLS method.