This work investigates a crucial aspect for the integration of MT technology into a CAT environment, that is the ability of MT systems to adapt from the user feedback. In particular, we consider the scenario of an MT system tuned for a specific translation project that after each day of work adapts from the post-edited translations created by the user. We apply and compare different state-of-the-art adaptation methods on post-edited translations generated by two professionals during two days of work with a CAT tool embedding MT suggestions. Both translators worked at the same legal document from English into Italian and German, respectively. Although exactly the same amount of translations was available each day for each language , the application of the same adaptation methods resulted in quite different outcomes. This suggests that adaptation strategies should not be applied blindly, but rather taking into account language specific issues, such as data sparsity.