We study the multi-level bottleneck assignment problem: given a weight matrix, the task is to rearrange entries in each column such that the maximum sum of values in each row is as small as possible. We analyze the complexity of this problem in a generalized setting, where a graph models restrictions how values in columns can be permuted. We present a lower bound on its approximability by giving a non-trivial gap reduction from three-dimensional matching to the multi-level bottleneck assignment problem. We present new integer programming formulations and consider the impact of graph density on problem hardness in numerical experiments.