The Lyα damping wing is seen in absorption against the spectra of high-redshift galaxies. Modeling of this wing is a way to measure the volume neutral hydrogen fraction of the Universe, as well as to measure the strength of H i absorption in the form of a damped Lyα absorber (DLA). We use very-high-quality ultraviolet spectra of low-z galaxies to create mock high-redshift, low-resolution spectra complete with a damping wing resulting from the intergalactic medium (IGM) and H i absorption from local, dense gas to mimic current James Webb Space Telescope observations of the early Universe. The simulated spectra are used to test the ability to recover the “true” values of the column density of the DLA (NDLA) and the volume neutral hydrogen fraction (xH I) of the IGM through modified stellar continuum fitting. We find that the ability to recover the true values of NDLA and xH I simultaneously is compromised by degeneracies between these two parameters, and the spectral shape itself at low resolution (R ∼ 100). The root mean squared error between the recovered and true column density of the DLA is on the order of 1 dex. This error somewhat decreases with improved resolution (R ∼ 1000), but systematically underestimates NDLA when Lyα emission is present in the mock spectra. The recovered values of xH I are poorly constrained and do not improve substantially with the higher resolution. We recommend accounting for these sources of uncertainty and biases when using galaxies’ Lyα damping wings to measure the intergalactic neutral hydrogen fraction.