Data sparsity
One of the biggest problems is that mostly the cube is very sparsely populated.
Many of the cell combinations might not make sense or the data for them might be missing. In the relational world storage of such data is not a problem: we only keep whatever there is. If we want to keep closer to our multidimensional view of the world, we face a dilemma: either store empty space or create an index to keep track of the nonempty cells. Or - search for an alternative solution.