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.
Also why do I want to stay with the cheap people? They proved to be unable to deliver market rates for my efforts unless I make a big scene about it. I don't like to do business with people like that.
It's no different than price shopping between mechanics. You can do that while being clear on your intentions and not making any promises you can't keep
I also routinely get asked by managers for promo recommendations for coworkers thinking about leaving. IF they refuse to counter, then you should be asking yourself why you are staying.
You cant pull this maneuver more than once or twice, but it lets them know you are serious and wont just whine and accept token raises in future negotiations.