The Birnbaum-Saunders (BS) distribution was introduced to model the degradation time of materials that are subject to wear. Recently, generalizations of this distribution have been introduced. With these generalizations, distributions with different degrees of skewness and different unimodal and bimodal types have been obtained. In this paper, generalizations of the BS distribution based on the level elliptic and level elliptic hump distributions are studied, which have high flexibility in terms of skewness and symmetry and are suitable for fitting to various types of cumulative damage data. For this purpose, first the level elliptic distributions are introduced in the univariate case and special cases of this family of distributions are stated. Then, we introduce the level elliptic hump distributions and then generalizations of the generalized BS distribution based on the level elliptic and level elliptic hump distributions are introduced. The specific properties of these distributions are outlined and examples of the application of these generalizations are provided.