A "packing" optimisation problem with application in dynamic programming
Yes.  This can be solved in polynomial time using a straightforward dynamic programming algorithm.  You build a table $A[\cdot,\cdot]$ where $A[i,m]$ denotes the maximal value of $f_1(n_1)+\dots+f_1(n_i)$ subject to the requirement that $n_1,\dots,n_i$ be a partition of $m$, i.e., $n_1 + \dots + n_i = m$.  The entries satisfy the following recursive relation:

$$A[i,m] = \max \{A[i-1,m-t] + f_i(t) : 0 \le t \le m\}.$$

You can fill in this array using the recursive relation in $O(kn^2)$ time, which is polynomial in $n$.  Then, $A[k,n]$ contains the answer to your problem.