We critically review the current null results on a varying fine-structure constant, α, derived from Very Large Telescope (VLT)/Ultraviolet and Visual Echelle Spectrograph (UVES) quasar absorption spectra, focusing primarily on the many-multiplet analysis of 23 absorbers from which Chand et al. reported a weighted mean relative variation of Δα/α= (−0.06 ± 0.06) × 10−5. Our analysis of the same reduced data, using the same fits to the absorption profiles, yields very different individual Δα/α values with uncertainties typically larger by a factor of ∼3. We attribute the discrepancies to flawed parameter estimation techniques in the original analysis and demonstrate that the original Δα/α values were strongly biased towards zero. Were those flaws not present, the input data and spectra should have given a weighted mean of Δα/α= (−0.44 ± 0.16) × 10−5. Although this new value does reflect the input spectra and fits (unchanged from the original work – only our analysis is different), we do not claim that it supports previous Keck/High Resolution Echelle Spectrograph (HIRES) evidence for a varying α: there remains significant scatter in the individual Δα/α values which may stem from the overly simplistic profile fits in the original work. Allowing for such additional, unknown random errors by increasing the uncertainties on Δα/α to match the scatter provides a more conservative weighted mean, Δα/α= (−0.64 ± 0.36) × 10−5. We highlight similar problems in other current UVES constraints on varying α and argue that comparison with previous Keck/HIRES results is premature.