All GFCI outlets have one little-known flaw: their circuitry eventually wears out, usually after about 10 years, at which point they no longer function properly. Here's the whole story.
The reset button alone won't tell you if a pre-2006 GFCI outlet is still working properly—you'll need to check it with a special tester.
By detecting dangerous current flow and instantly shutting off power, ground fault circuit interrupters save hundreds of lives each year. But after 10 years or so, the sensitive circuitry inside a GFCI wears out. And usually the test button on the GFCI doesn't tell you there's anything wrong: When you press the button, it shuts off the power as always. So the only reliable way to check an older GFCI is to use a circuit tester that has its own GFCI test button (sold at home centers and hardware stores).
Plug in the tester and push its test button. If the power goes off, the GFCI is working. Press the reset button to restore power. If the power doesn't go off, replace the GFCI.
Your new GFCI will never require a circuit tester. All GFCIs manufactured after mid-2006 are designed to tell you when they fail. The vast majority indicate failure by shutting off power permanently. So someday your GFCI (and any other outlets connected to it) will simply stop delivering power and you'll have to replace it.