For cocycles defined by sequences of linear operators in a Banach space, we study the relation between the notions of nonuniform exponential stability and admissibility. The latter refers to the existence of bounded solutions for any bounded nonlinear perturbation of the original cocycle. In particular, using appropriate adapted norms to deal with the nonuniform behaviour, we show that if is admissible for a given cocycle, then the cocycle is a nonuniform exponential contraction. We also exhibit a collection of admissible Banach spaces for any given nonuniform exponential contraction. In addition, we obtain related results in the more general case of nonuniform exponential dichotomies. As an application of the ideas and methods discussed in the paper, we establish the robustness of nonuniform exponential contractions, in the sense that a sufficiently small linear perturbation of a nonuniform exponential contraction is again a nonuniform exponential contraction.