You mention additive number theory in the question, so perhaps this isn't the type of example that you want. However my understanding is that the Three Primes Conjecture (every odd number $\geq 7$ is the sum of three primes) is now at the cusp of a feasible computer solution.
Vinogradov proved that the conjecture was true for all sufficiently large odd numbers (i.e. there exists $C>0$ such that every odd number greater than $C$ is the sum of three primes).
Various people have given explicit values for $C$ but, up until recently, the best (i.e. lowest) explicit value was $e^{3100}$. This is still, obviously, way out of computational range.
However Tao and, then, more recently Helfgott have improved these bounds by studying so-called `minor arcs', so that now one could just about imagine dealing with remaining cases via computer. The results of Helfgott are expressed in terms of a parameter $q$ pertaining to minor arcs. His work implies that the conjecture is true for $q>4\cdot 10^6$.
If you're interested you should read Helfgott's preprint which begins with a brief summary of the history of this problem.
