swap another derailleur on... see if that fixes the issue.... sounds like the spring is shot... they DO wear out! have you LUBED the derailleur yet? they get sticky, or can rust. Is it an old bike/derailleur? springs do wear out... they can only flex so many times before metal fatigue sets in...
a worn roller would make shifting baulky and imprecise, but it WOULD still shift.
heck, test the tension of the return spring... do this by feel, or use a big rubber band, note the length when the derailleur moves then test a known, good working derailleur... you WILL need to remove the rear wheel to do these tests.