You can make a cube out of 27 cubes of the same size like this:

But can you also make a cube out of 27 cubes that
**
aren't
**
all the same size?

**
Assumption:
**
Each cube has a finite size.

No
Yes

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Take a 3x3x3 cube. Combine 8 of them that are in a 2x2x2 cube leaving 20 cubes. Then divide one of the original ones up to make a new 2x2x2 cube with smaller cubes. The cube is now made up of 27 cubes of 3 different sizes.

So, $\boxed{\text{yes}}$ you can make a cube using 27 cubes without all of them being the same size.