I had 2 separate instances like this on ours: One was corrosion inside the starter cable. Water had seeped past the end lugs, and got inside the copper. After much diagnosis, my mechanic jumpered around that cable, and it worked fine. Once we removed the starter cable and cut the insulation, you could see the corrosion. From the outside view, the cable looked fine.
2nd instance was the transmission linkage...The car would not fully engage Park, so it wouldn't start. It would hold the car fine, but not engage enough to hit the safety switch...eventually I found out I could start in in Neutral.
But I'm in Canada, corrosion is a pain on any car.