We consider the non-linear optimal multiple stopping problem under general conditions on the non-linear evaluation operators, which might depend on two time indices: the time of evaluation/assessment and the horizon (when the reward or loss is incurred). We do not assume convexity/concavity or cash-invariance. We focus on the case where the agent's stopping strategies are what we call Bermudan stopping strategies, a framework which can be seen as lying between the discrete and the continuous time. We first study the non-linear double optimal stopping problem by using a reduction approach. We provide a necessary and a sufficient condition for optimal pairs, and a result on existence of optimal pairs. We then generalize the results to the non-linear $d$-optimal stopping problem. We treat the symmetric case (of additive and multiplicative reward families) as examples.