New York Times Archive Free
The New York Times now offers online access to the full text of articles from 1851 to 1922, and from 1987 to the present. From the New York Times homepage at http://www.nytimes.com/, simply enter a search in the box at the top of the page, and in the dropdown box to the right select the time period you'd like to search. This service is free to all.
Milne Library offers access to the full content of the New York Times from 1851 to 2004 to all SUNY Oneonta users. Look for the link under Databases by Name (A-Z) on the library home page. For 2004 to the present, use Lexis-Nexis Academic Universe. The advantage of the library access is that the 1923 to 1986 years are available, and that you can conduct a much more sophisticated search.