A problem similar to bin packing
Your problem belongs to the family of problems known as fixed interval scheduling. For a recent survey, see Kovalyov et al. A classical paper of Arkin and Silverberg solves the problem when all sizes or capacities are the same, and shows that if you generalize the notion of size then the problem of deciding whether all jobs (i.e., items) can be scheduled (i.e., stored) is NP-complete, though it has an $O(n^{k+1})$ solution, where $n$ is the number of jobs and $k$ is the number of machines (i.e., containers).

Your problem is easier than the NP-complete variant considered by Arkin and Silverberg:


Your constraints on which jobs can be scheduled on which machines are not arbitrary – they depend on the notion of size.
Your timeframe is short.
You are guaranteed a solution. (Though this doesn't usually make the problem any easier.)


The literature on the problem is probably vast, so perhaps someone already considered your particular variant.