The impact of input errors in the calibration of watershed models is a recurrent theme in the water science literature. It is now acknowledged that hydrological models are sensitive to errors in the measures of precipitation and that those errors bias the model parameters estimated via the standard least squares (SLS) approach. This paper presents a Bayesian uncertainty framework allowing one to account for input, output, and structural (model) uncertainties in the calibration of a model. Using this framework, we study the impact of input uncertainty on the parameters of the hydrological model "abc." Mostly of academic interest, the "abc" model has a response linear to its input, allowing the closed form integration of nuisance variables under proper assumptions. Using those analytical solutions to compute the posterior density of the model parameters, some interesting observations can be made about their sensitivity to input errors. We provide an explanation for the bias identified in the SLS approach and show that in the input error context the prior on the input "true" value has a significant influence on the parameters' posterior density. Overall, the parameters obtained from the Bayesian method are more accurate, and the uncertainty over them is more realistic than with SLS. This method, however, is specific to linear models, while most hydrological models display strong nonlinearities. Further research is thus needed to demonstrate the applicability of the uncertainty framework to commonly used hydrological models.