It's "business." Your manager isn't going to make a counteroffer, then wait a month and fire you. I mean, they might if they're a total D.bag, but I've not yet had that experience. And if I had, I'd already proven to myself and my manager that I could get a higher-paying job elsewhere. It'd be a minor annoyance on my side.
I don't want to give away specifics, so take these numbers with a lump of salt; the ratios are correct:
Went from $50k/yr [1]-> $80k/yr [2]-> $120k/yr [3]-> $205k/yr
[1] The company wanted me to stay and raised my pay to match the competing offers.
[2] The company wanted me to stay and raised my pay to match the competing offers.
[3] The company didn't want to pay that much, so I put in my two weeks and moved to a different company.
From my experience, I've found that requesting a raise without a competing offer often leads to a small or no increase in pay. However, when I've approached my manager with a better offer in hand, it has consistently resulted in substantial pay raises.