A proper uncertainty assessment of rainfall-runoff predictions has always been an important objective for modelers. Several sources of uncertainty have been identified, but their representation was limited to complicated mechanistic error propagation frameworks only. The typical statistical error models used in the modeling practice still build on outdated and invalidated assumptions like the independence and homoscedasticity of model residuals and thus result in wrong uncertainty estimates. The primary reason for the popularity of the traditional faulty methods is the enormous computational requirement of full Bayesian error propagation frameworks. We introduce a statistical error model that can account for the effect of various uncertainty sources present in conceptual rainfall-runoff modeling studies and at the same time has limited computational demand. We split the model residuals into three different components: a random noise term and two bias processes with different response characteristics. The effects of the input uncertainty are simulated with a stochastic linearized rainfall-runoff model. While the description of model bias with Bayesian statistics cannot directly help to improve on the model's deficiencies, it is still beneficial to get realistic estimates on the overall predictive uncertainty and to rank the importance of different uncertainty sources. This feature is particularly important if the error sources cannot be addressed individually, but it is also relevant for the description of remaining bias when input and structural errors are considered explicitly.