We revisit classical asymptotics when testing for a structural break in linear regression models by obtaining the limit theory of residual-based and Wald-type processes. First, we establish the Brownian bridge limiting distribution of these test statistics. Second, we study the asymptotic behaviour of the partial-sum processes in nonstationary (linear) time series regression models. Although, the particular comparisons of these two different modelling environments is done from the perspective of the partial-sum processes, it emphasizes that the presence of nuisance parameters can change the asymptotic behaviour of the functionals under consideration. Simulation experiments verify size distortions when testing for a break in nonstationary time series regressions which indicates that the Brownian bridge limit cannot provide a suitable asymptotic approximation in this case. Further research is required to establish the cause of size distortions under the null hypothesis of parameter stability.