As a rule I've never entertained counters. I'm sure that management always keeps that kind of thing in the back of their mind, whether it influences anything probably depends on where you are. But, more importantly, if I went to the pain in the ass to update my resume and do a job search and interviews and all that stuff, that means I was pretty unhappy where I was at for some reason. The way I am, I will have tried to fix those things that made me unhappy before I bailed, so if I've gone through the process of trying to fix the stuff and I've also gone through the asspain of a job search, and only now when I'm heading out the door do I get traction on whatever the issues where, yeah that's a no from me.
Last job I left countered me hard, it was a significant bump. It still wasn't hard to turn down, because that place wasn't great, the on site manager was a blowhard who knew about 30% of what he pitched himself as, and I would have just stagnated and died there.