“The problems of the world cannot possibly be solved by skeptics or cynics whose horizons are limited by the obvious realities. We need men who can dream of things that never were and ask "why not?"”

Though Kennedy stated that he was quoting George Bernard Shaw when he said this, he is often thought to have originated the expression, which actually paraphrases a line delivered by the Serpent in Shaw's play Back To Methuselah: “You see things; and you say, ‘Why?’ But I dream things that never were; and I say, ‘Why not?’". This phrase was first used by his brother John F. Kennedy in 1963 (June 28th), during his visit to Ireland, in his address to the Irish Dail (Government): "George Bernard Shaw, speaking as an Irishman, summed up an approach to life, 'Other people, he said, see things and say why? But I dream things that never were and I say, why not?" ( Address on YouTube https://www.youtube.com/watch?v=8ADeazX9blw.). Robert's other brother Edward famously quoted it (paraphrasing it even further), to conclude his eulogy to his late brother after his assassination (8 June 1968): Some men see things as they are and say why? I dream things that never were and say why not? - (Eulogy in CBS news video) http://www.cbsnews.com/video/watch/?id=5268061n
Misattributed
Source: Robert Kennedy in His Own Words: The Unpublished Recollections of the Kennedy Years

Ibid., p. 282
The Book of Disquiet
Original: Como nunca podemos conhecer todos os elementos de uma questão, nunca a podemos resolver.

Speech at Catholic University, Columbus School of Law http://web.archive.org/web/20040704015129/http://www.law.cua.edu/News/Things%20That%20Never%20Were.cfm (2004).
2000s

Attributed

Cf. Bernard Shaw, Back to Methuselah (1921), Pt. I : In the Beginning: I hear you say "Why?" Always "Why?" You see things; and you say "Why?" But I dream things that never were; and I say "Why not?"
Books, Brain Droppings (1997)
Variant: Some people dream of things that never were and ask, Why not? Some people have to go to work and don't have time for all that shit.

Letter to Robert McAlmon (4 September 1943), published in The Selected Letters of William Carlos Williams (1957) edited by John C. Thirlwall, p. 217
General sources
Context: Why do we live? Most of us need the very thing we never ask for. We talk about revolution as if it was peanuts. What we need is some frank thinking and a few revolutions in our own guts; to hell with what most of the sons of bitches that I know and myself along with them if I don't take hold of myself and turn about when I need to — or go ahead further if that's the game.