Solution:
Divide all the cubes into unit cubes. Then the 1996 cubes must each contain at least one unit cube, so the large cube contains at least 1996 unit cubes. But $12^3 = 1728 < 1996 < 2197 = 13^3$, so it is certainly not possible for $n < 13$.

It can be achieved with 13 by $1 \cdot 5^3 + 11 \cdot 2^3 + 1984 \cdot 1^3 = 13^3$ (actually packing the cubes together to form a $13 \times 13 \times 13$ cube is trivial since there are so many unit cubes).