In part 2 of the episode "In the shadow of 2 gunmen", the aftermath of the assassination attempt look for the flash-back were Josh is trying to call Sam from a pay phone after seeing Bartlet in New Hampshire. The pay phone has a HUGE Verizon sticker on it, 3 years before the company was formed.
Nice catch. Reminds of the famous slip in Die Hard II--where Bruce Willis makes a call on the East Coast with a phone booth with Pacific Bell plastered on it.
Actually, it has been established that the show takes place in the present, not the near-future. I wish I could recall more details, but there was an episode in late 2000 in which Sam, in one of his usual "I'm-hung-up-on-something-for-the-entire-episode" bits, keeps nattering on about when the millennium actually ends and how most people think it's going to end on New Year's Eve and so forth. The key being that he's talking about it as a current event.
If this is the case, the Verizon product placement was a big slip-up. I remember watching this episode and going "Oops!" when I saw it. Nice catch.