The article is not especially pay-walled, from what I can detect, using multiple devices. It seems to just use the typical 10 article cookie variable that all NYT articles usually check.

I'd estimate your current machine is behind some kind of network/web proxy that's browsing with a separate session, on your behalf, with a stale cookie that has reached the free article limit attached to its session.

Maybe you're the victim of an A/B test? That's the only other idea I can imagine.