Weather forecasting is challenging due to the exceptional complexity of the atmospheric phenomena involved. Modern weather forecasts are typically in the form of an ensemble of forecasts obtained from multiple runs of numerical weather prediction models. Ensemble forecasts are often biased and affected by dispersion errors, and they should be statistically corrected to gain accuracy. The standard correction methods, such as Ensemble Model Output Statistics (EMOS), only apply to a single variable of the forecasting problem at a time. This results in a loss of the dependence structure of the multivariate forecasts, which is problematic in several applications. Recent work shows that the lost dependence structure can be efficiently reconstructed via non-parametric multivariate post-processing approaches based on empirical copulas. Popular methods of this type are Schaake Shuffle, Ensemble Copula Coupling (ECC), and SimSchaake. In this work, we inquire into the limitations of the empirical copula methods for the statistical correction of temperature forecasts of the Austrian ensemble system ALADIN-LAEF. Our setting is challenging: ALADIN-LAEF has been running operationally for six years only, and, in general, it might not be considered fully exchangeable. Given these issues which affect ECC and SimSchaake in different ways, a natural question arises whether or not these multivariate modeling approaches are still effective. In this paper, we present a case study aiming at answering this question. We consider three groups of stations with different characteristics: three close stations in a valley, three stations on top of mountains, and three randomly chosen stations within a distance of several hundreds of kilometers. For each group of stations, we compare the performance of SimSchaake, ECC, and individual EMOS to correct the multivariate temperature forecasts. Our analysis suggests that the non-exchangeability of the ALADIN-LAEF system is not a limitation to applying ECC to this ensemble system. Though, our results show that SimSchaake outperforms individual EMOS and ECC in all cases, supporting the claim that even in such an unfavorable scenario we can benefit from the use of empirical copula methods.