The elastic nucleon-deuteron scattering process can be used to study three-nucleon dynamics, but in order to draw reliable conclusions based on the comparison of theoretical predictions and data, an estimation of theoretical uncertainties is necessary. We consider the statistical errors arising from a propagation of uncertainties of the two-nucleon interaction parameters to the three-nucleon system, truncation errors, regulator dependencies, and uncertainties arising from using the formalism of partial waves. We give an example of the nucleon-deuteron spin correlation coefficient Cz,x, obtained within the Faddeev approach, to discuss magnitudes of the above mentioned theoretical uncertainties. We find that using various models of the nuclear forces results in the dominant theoretical uncertainty for the nucleon-deuteron elastic scattering observables.