So, stiffer spine = heavier arrow (assuming same arrow). That could be a good thing or a bad thing depending on what you're looking for. Heavier shafted arrow for the same spine makes it harder to get higher FOC, you may or may not care about that.
Personally, I've never had an issue going too stiff in terms of tuning and arrow flight, that's what TBone and Aron were talking about. Especially if I make sure it's a good straight arrow, spins correctly, and my bow is tuned well. If you're talking about jumping up from a 340 to a 300, you won't have an issue.
With that said, People like Dudley do talk about the "HIL Method" or other group tuning of different spined arrows, and picking the spine that gives you the best groups. And I've heard people say that the "right" spine can give you better grouping than one that's too stiff or too weak. I just personally haven't done enough testing to see it, and usually try to stay in the good range to a little stiff and the arrows and bow are always more accurate than I am.