Michael Lugo's argument can be formalized and generalized. The sequence $(a_j)_{j \geq0}$ is {\it log concave} (also known as strongly unimodal) if $a_{j-1}/a_{j} \leq a_{j} /a_{j+1}$ for all relevant indices $j$. The binomial sequence is log concave. For $k$ less than and far away from the location of the maximum, there is a simple geometric series argument. Specifically, if $\rho = a_ {k-1}/a_{k} $, this is less than $1$, and so for $j<k$, the term $a_j \leq \rho^{k-j} a_k$. Hence$$\sum_{i=0}^k a_j \leq a_k \sum_{i=0}^{k-1} \rho^i \leq \frac{a_k}{1-\rho}.$$The last inequality is not very effective when $\rho$ is very near $1$, and in that case, there is usually a better approximation available (since $\rho^i$ doesn't change much over a long range ...; the details are usually relatively easy to work out). For $\rho$ noticeably smaller than $1$, the upper bound so obtained can be refined because the actual sequence of ratios of consecutive binomial coefficients decays quickly, so in general only a few terms of the geometric series are needed.
This has the advantage of working for all log concave sequences, for example, the reparameterized sequence $(t^j a_j)$ for $t > 0$, and it is a starting point for approximations nearer the location of the maximum.
