The wetting of polymer brushes exhibits a much richer phenomenology than wetting of normal solid substrates. These brushes allow for three wetting states, which are partial wetting, complete wetting and mixing. Here, we study the transitions between these wetting states for brushes in contact with polymer melts and compare them to predictions using enthalpic arguments based on brush and melt interactions. We show that the transitions are shifted compared to the enthalpic predictions and that the shifts can be positive or negative depending on the length of the melt polymer and the brush grafting density. The reason for this is that these brush and melt parameters can have a positive or negative effect on the entropic contribution to the free energy of the system. Our results highlight the relevance of entropy in predicting the exact wetting transitions, which is important for the design of brush-based coating applications.