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Uncertainty propagates through calculations, down to molecular scales to infer microstructural features, and up to macroscopic scales with predictive flow simulations. Here we study uncertainty quantification for sequential (two-step) versus simultaneous (all at once) fitting methods with linear and weakly-nonlinear rheological data. Using an example of a combined dataset on small-amplitude oscillatory shear (SAOS) and medium-amplitude oscillatory shear (MAOS) for a linear entangled polymer melt (cis-1,4-polyisoprene), we demonstrate with a multi-mode Giesekus model how the fit parameter uncertainties are significantly under-estimated with the sequential fit because of the neglect of model parameter correlations. These results are surprising because weakly-nonlinear data is only an asymptotic step away from the linear data, yet it has significant impact on calibrating the linear model parameters. Similarly, the nonlinear parameter estimates and uncertainties are impacted by considering the linear data in a simultaneous fit. To compare multi-mode spectra of nonlinear parameters (mobility parameters {α_i} from the Giesekus model), we derive new average measures based on moments of the spectra related to the high-frequency MAOS limit. The spectral averages are also sensitive to sequential versus simultaneous fitting. Our results reveal the importance of using simultaneous fitting for honest uncertainty quantification, even with weakly-nonlinear data.