Answer by Douglas Zare for Sum of 'the first k' binomial coefficients for fixed $N$

One standard estimate when the sum includes about half of the terms is the Chernoff bound, one form of which gives

$$\sum_{k=0}^{(N-a)/2} {N\choose k} \le 2^N \exp\bigg(\frac{-a^2}{2N}\bigg)$$

This isn't so sharp. It's weaker than the geometric series bound Michael Lugo gave. However, the simpler form can be useful.