Actually this seems like an interesting question to me. One can easily calculate the maximum number of regions obtained by n hyperplanes:

For lines in $\mathbb{R}^2$, by induction, the maximum number of regions achievable with $n$ lines is $1+1+2+ \ldots + n$. For planes in $\mathbb{R}^3$, denote the maximum regions by $N_n$. Then one sees that $N_{n+1} - N_n = $ the maximum number of regions in $\mathbb{R}^2$ achievable by $n$ lines, hence equals $1+2+ \ldots + n$. Thus $N_n = 1+ n + (n-1) + 2(n-2) + 3(n-3) + \ldots (n-1)$.

The in-between numbers seem much more elusive. Even the version of the problem for lines in $\mathbb{R}^2$ seems hard. I found by experimenting that 5 is not achievable by any number of lines in $\mathbb{R}^2$ less than 4. So a natural question could be what number $n$ has the property of not being achievable by any number of lines less than $n-1$.

For the special case of 4 planes in $\mathbb{R}^3$. I think the correct answer is: 5, 8, 9, 10, 11, 12, 14, 15. It's clear 6,7 aren't constructible. 13 is not constructible by brute force checking all constructible numbers with 3 planes and seeing that it's impossible to add another plane to get 13.