dimension of intersection of hyperplanes

By hyperplane I assume you mean an affine space of dimension $n-1$, which can be defined by a single linear equation in $n$ variables.

In general, when a hyperplane intersects an affine space of dimension $m$ there are three possibilities:

1. The hyperplane includes the affine space, so the intersection still has dimension $m$.
2. The hyperplane is parallel to the affine space, so the intersection is empty.
3. (The usual case) There is a proper intersection, and the intersection has dimension $m-1$, one less than the affine space.

In summary, each hyperplane leaves the number of dimensions unchanged or reduced by one, or makes the result empty (with an undefined number of dimensions).

So if $k>=n$ the intersection can have any number of dimensions from $0$ to $n-1$ (as Linus S. said in a comment) or empty.

If $k<n$ then the intersection can have any number of dimensions from $n-k$ to $n-1$, or empty. It is not possible to get $0$ through $n-k-1$ dimensions or $n$ dimensions. For example, in three dimensions you cannot get two planes to get the intersection of a point ($0$ dimensions) or all of space ($3$ dimensions): you get a line ($1$ dimension, if the planes intersect), a plane ($2$ dimensions, if the planes are identical), or nothing (empty, if the planes are parallel).

You could summarize this by saying that the intersection has $\max(n-k,0)$ through $n-1$ dimensions or is empty, and empty can occur only if $k>1$.