Can't speak to the NYT, but I long ago dropped daily delivery of the local rag, The Columbus Dispatch. For a while we just got Sundays, then they forced Saturday on us (no Sunday only). Saturdays paper goes into the trash unopened, and these days likely as not so does Sunday's. The ONLY draw is the weekly advertising and funnies (guilty), and if I don't get to them within a day or two even they aren't worth taking the thing out of the plastic for.
News? What news? Every national story printed is two to three days old, because I've already heard about them here and elsewhere on the net. Local stuff I hear about on the radio, or in the free rags that arrive on the doorstep. Local TV news is in the same boat as the Dispatch as far as I am concerned.
I did that for a while but now will not even read the front page of our paper in the grocery store. For a while they kept calling me trying to get me to subscribe again and I would say stuff like, "You couldn't pay ME to take your inferior and ignorant collection of stories."
Mine is one of the worst, The Atlanta Journal Constitution (AKA Al-Jazeera Constipation).