I've looked and looked and looked, and, as I expected, nobody was actually clever enough to ask or test this out. Everybody seems to be saying that it "adds Flinch chance", not "raises Flinch Chance", so based on that, here's a possible theory:

It's kinda like stacking Infatuation with Paralysis. Both have a 50% chance of making the infected Pokemon not attack, but they're not stacked for 100% immobility. You'll have a 50% chance of breaking through Infatuation. If you break through it, you have to chance another 50/50 shot at resisting paralysis.

Stench is the same way. Used with Astonish, you have to calculate two separate Flinch chances: One for calculating Stench's Flinch chance, and another for Astonish's. So in the end, you won't have a 20% flinch rate, but rather, two 10% Flinch Rates; not exactly the favorable outcome....

This is purely a guessing game based on what people have said about the ability itself, though. Wait for other answers, and I'll keep looking. If I find confirmation on what's really happening, I'll edit my answer.