Why does everyone tell me that I shouldn't accept an increase in money to stay? Sometimes companies need a push.

Generally speaking, a counter offer is usually based around a pay increase.  And you have to ask yourself "is $XXXX dollars going to solve the issues I have with this company or my job?"

If it does solve the issue, then great.  But if you take a 5 -10% increase and you hate your boss, do you think he/she will change their personality?  Probably not.  But if it is a commute issue, and that 5-10% covers the time lost in the car or on the train, then that increase can make the difference.  Although, wouldn't it have been easier just to ask for a raise?
 

Those that accept counter-offers at the professional level really do shoot themselves in the foot for many of the reasons mentioned above.  The most successful people in business are risk takers & solid decision makers.  Someone who accepts a counter offer has shown that they are not willing to take a little risk by leaving what they know to venture into something new, and they have now gone back on their word and have waffled, so they are no longer viewed as a solid decision maker.  So now, instead of moving on to potentially bigger and better things, they have done the complete opposite.  I am sure it does pan out for some people, I would be very interested in learning about how the acceptance of a counter-offer has indeed worked out for someone's career.