The UCLA Library does not subscribe to latimes.com, nytimes.com, or any other newspaper websites. In fact, most newspaper sites don't even have an option for institutional subscriptions.
Instead we subscribe to third-party newspaper databases which license the articles from the newspaper publishers. These archives cover long time periods, provide advanced search features typically available in academic databases, and don't have ads! (Though they often index the ads present in the print newspapers, allowing you to actually search them.) Current archives are usually updated daily.
On the down side, archives are primarily about storing text and (sometimes) images. They usually don't contain videos or interactive features available on the original news sites.
http://archive.org - online versions of microfilm reels; search term: UCLA Daily Bruin. Keyword searching is available within each reel
The Daily Bruin went online in Fall Quarter, 1994, http://dailybruin.com/