Do I need to play any of the previous games in the Fallout series prior to Fallout 4?

Is Fallout 4 reasonably self-contained, or are there important pieces of backstory that would be better understood by playing earlier titles in the series?

There are nods to other Fallout games, and the game takes place in the same universe, but you will be able to enjoy Fallout 4 without having played any of the previous ones. The introduction of the game explains the premise quite well and gets you up to speed quickly.

I think the main thing you need to take from the lore in my opinion is the war of 23 October 2077 - The day of the nuclear holocaust, and the resource wars between the US and China that led to it (China being often referred to as the "communist threat"). The theme of the game is that of a futuristic 1950's due to how the timeline of fallout diverges from real history after the second world war. Instead of microprocessors, the world chose to pursue nuclear research, hence the nuclear themed world (cars using nuclear fission and even drinks!).

You will not be lost at all if you did not play the previous games. Those of us who did play them all will get a little extra kick out of the game when we are faced with the references and the throwbacks to some of the older games. Having said all that... do not worry, you will enjoy the world of Fallout and if you get sucked in you can always visit the older titles at a later date if you wish (which I can highly recommend - but it is my biased opinion).