A familiar example is the melting of ice. Here is a change that we know is endothermic and occurs with an increase in entropy. At lower temperatures, ice doesn't melt because the smaller value of T gives a smaller value for TS and the difference H-TS, is positive. For similar reasons, when H and S are both negative, G will be negative only at relatively low temperatures. The freezing of water is an example.
Energy is released as the solid is formed and the entropy decreases.