NYT dialed back their free access at the beginning of April:
"Visitors can enjoy 10 free articles (including blog posts, slide shows, videos and other multimedia features) each calendar month on NYTimes.com, as well as unrestricted access to browse the home page, section fronts, blog fronts and classifieds.
"Your free, limited access resets every month: at the beginning of each calendar month, you'll once again be able to view 10 free articles for that month."

The toll-free telephone number seems to be legit; it's the same one given on their website for Customer Service.