Better make sure you have that other offer in writing if you do.
This.
OP, I'm torn on your question to be honest. I have friends who were managers and they always told me that in their opinion, you should always go to your manager and have a discussion if you're unhappy with your pay, role, career path, etc. One friend remarked that he had employees leave who told them they were unhappy and he literally had no clue that they were unhappy and would have made things better if they had told him. Of course, this friend was a great guy with no political agendas whatsoever, so I think he is in the extreme minority and perhaps a bit naive. He was a great boss, though.
On the other hand, my manager at my last job told me never to accept a counteroffer if given one and that he would never extend one if an employee quit. According to him, statistics show that an employee who accepts a counter will still be unhappy and leave within a couple of years. Furthermore, from the perspective of the employer, you've already proven that you are testing the waters and aren't loyal, so they may start looking for your replacement and get rid of you at the next opportunity they have (such as layoffs).
Personally, I don't discuss these kinds of things with my managers. I feel that if they can't see if I am overworked or underpaid, they're failing at their job as a manager. What I typically do is set a course of action, achieve it without their knowledge (such as training for the PMP), and then find another job. Realistically speaking, if you have no upward mobility at your current employer, you're going to have to leave to get a significant raise.