Yes, you can access full text of The New York Times articles online via the Nexis Uni database, from 6/1/1980-present, as well as within Gale's databases such as New York State Newspapers (1985-present).

You can find which databases to use for NYT and many other newspapers/magazine/journals using the Periodical Finder in Primo Search.

Note: If you are having trouble linking to full text in various databases from Primo Search results, you can try searching directly within the indicated databases to access specific articles.  [See the A-Z Databases list.]