GCFI "test" button will not cause a trip
This can happen, just replace the GFCI outlets in question

The TEST button on a GFCI creates an actual ground fault (about 6-8mA, maybe up to 10mA, from load-hot to line-neutral or vice versa), so if a GFCI fails to trip when the TEST button is pressed, then it's dead and needs to be replaced.  This can happen due to improper wiring (backfeeding a GFCI's LOAD terminals will burn out the trip solenoid), or more commonly due to damage to the GFCI's (largely analog!!) electronics package caused by mains surges and spikes.  (More modern GFCI designs may have a varistor connected across line-hot and line-neutral to help protect the electronics from surges, but this is by no means guaranteed.)