Back-of-napkin-math time... a typical shipping container is 1,360 cubic feet, that's 2,350,080 cubic inches. A bag of paws/simple/space is about 1,024 cubic inches, so that's about 2,295 bags per container, or at 8 bags per case, 286 full cases per container. (somehow I thought those containers held more? but then again your cases are pretty big!) (I'm sure my precision isn't very good having to work my way up from inches though)
If you sold "thousands" of cartons (cases?), you must be ordering several containers at a time now. Also if we looked at a commonly quoted "100,000 pc minimum order", at an estimated 18,360 diapers per container, that's five and a half containers per minimum order, so I suppose you have to order multiple containers at a time. (I was thinking a "minimum order" might fit into one container... WRONG!) Rounding up the minimum case order if we assume shipping charges by the container regardless of weight, six containers would hold just a bit shy of 2,000 cases of little paws. (all of this is assuming size Large... I'm not sure if med/small take up less volume per diaper but I think they do?)
And I suppose realistically speaking, a "warehouse" proper could hold quite a few shipping containers of product. (assuming you're not sharing the warehouse with some other company) Multiply that by the number of different products (and sizes) you carry, and it totally makes sense that you're tight on warehouse space after a few shipments come in. Here I thought you had two warehouses to cut down on shipping time or distance, when needing more space was probably the biggest reason!