Some recent results in state complexity were found with the help of systematic brute-force search for worst-case examples. This is doable because there are not too many deterministic finite automata with a small number of states, for example if we concentrate on binary or ternary alphabets. Also, in many cases there are families of worst-case examples for $1,2,3,4,5$ states and so on, and not only for all numbers of states $n$ that are greater than some large-ish $n_0$.
I've found this explicitly described in the introduction of the following paper.
Guangwu Liu, Carlos Martin-Vide, Arto Salomaa, Sheng Yu: State complexity of basic language operations combined with reversal. Information and Computation 206 (2008) 1178–1186
The relevant quote is as follows:
All the main results presented in this paper, especially the lower bound results, were obtained with the help of a large
number of experiments using computer software. For obtaining the tight lower bound for the reversal of union, for example,
we first proposed possible examples from our experiences. Then we checked the examples using our computer software
Grail+. There were many iterations in modifying the examples and checking by the software again. After satisfying examples
were obtained with experiments, we tried to prove the result theoretically. When we had problems in obtaining a theoretical
proof, we were back to experiments again. We consider that this is a relatively new approach in theoretical computer science
research. We think that our results presented in the paper would be very difficult to obtain without this new approach.
For many results in state complexity, the upper bounds are from standard constructions, and the difficult part is to find worst-case examples and prove the tight lower bound. Many of the previous papers on state complexity discussed the proofs and did not lay out how the examples were found. But I remember that in 2008, the above method already had been around for at least a few years, and was considered a trick of the trade.
By the way, I think that also many results in state complexity are and were obtained without the computer assisted method, but in the traditional way of doing mathematics.