A $(k,k-t)$-SCID (set of Subspaces with Constant Intersection Dimension) is a set of $k$-dimensional vector spaces that have pairwise intersections of dimension $k-t$. Let $\mathcal{C}=\{π_1,\ldots,π_n\}$ be a $(k,k-t)$-SCID. Define $S:=\langle π_1, \ldots, π_n \rangle$ and $I:=\langle π_i \cap π_j \mid 1 \leq i < j \leq n \rangle$. We establish several upper bounds for $\dim S + \dim I$ in different situations. We give a spectrum result for the case $(n-1)(k-t)\leq k$ and for the case $n\leq\frac{q^{t(n-η)}-1}{q^t-1}$, giving examples of $(k,k-t)$-SCIDs reaching a large interval of values for $\dim S + \dim I$.