Answer by Max Alekseyev for Sum of 'the first k' binomial coefficients for fixed $N$

Two bounds that work uniformly for all $k$:$$f(n,k) \leq \binom{n+1}{k} + \binom{n+k-1}{k-1},$$$$f(n,k)\leq \binom{n+k}{k},$$where the former is tighter for $k\geq 2$.

For a proof, see this answer.