The problem statement is somewhat unclear. I interpret it as follows: To every rectangular box, you assign the sum of its three dimensions as a measure of its "size". You then ask whether a box of greater "size" can be packed inside a box of lesser "size".
If you allow cutting up the box of greater "size", the answer is yes. For instance, a box with dimensions $3\times3\times3$ has "size" $9$ and volume $27$, whereas a box with dimensions $1\times4\times5$ has "size" $10$ and volume $20$, so you can cut the "bigger" one up, for instance into $20$ unit cubes, and pack it into the "smaller" one.
If cutting isn't allowed, this isn't possible. This is proved rather elegantly in Section 9 of this article. The basic idea is to consider the volume of space at a distance at most $\epsilon$ from the box. This has contributions in various powers of $\epsilon$, stemming from the various rounded bits surrounding the box, and the leading contribution that depends on the box dimensions is due to the quarter-cylinders parallel to each of the edges. This goes as $\epsilon^2$ times the sum of the box dimensions. Since for any given $\epsilon$ the inner box augmented by $\epsilon$ lies within the outer box augmented by $\epsilon$, this leading contribution can't be bigger for the inner box than for the outer box, and thus the "size" of the inner box can't exceed that of the outer box.