The Wikipedia article has a section on that:

Multiple knapsack problem

This variation is similar to the Bin Packing Problem. It differs from the Bin Packing Problem in that a subset of items can be selected, whereas, in the Bin Packing Problem, all items have to be packed to certain bins. The concept is that there are multiple knapsacks. This may seem like a trivial change, but it is not equivalent to adding to the capacity of the initial knapsack. This variation is used in many loading and scheduling problems in Operations Research and has a PTAS.

I'm afraid I'm not familiar with this variation, but having a name to search for should help, I reckon.

The OP surely wants to store the entire directory on his tapes. If so, MKP won't help. MKP is for when you have a fixed number of bins and stuff might not fit in the bins.

You're likely right. Thanks for enlightening me, brother!