A Northern Ireland version of that might, just might, have happened in Belfast last week. Somewhere along the line in the course of the week came a moment when it people began to think that success in the talks was more likely than not.
This must seem a pretty audacious observation, given that the week ended in setback when chairman George Mitchell was unable to produce a vital working paper; given that so many key points remain unresolved; and given that the prevailing mood can fluctuate dramatically, sometimes on an hourly basis.
Success is by no means assured, yet there is a sense in the air that some sort of corner was turned back there. This is not to assert that some new bonds of friendship and fellowship have been forged: it's not like that. This is Northern Ireland, and there remain vicious personal antagonisms, both between Unionism and nationalism and indeed within those two camps. If there was a change in the chemistry it is a political phenomenon, not a personal one.
Yet there are underlying relationships which - though often obscured - are among the important factors working towards agreement, and which will strongly underpin any new deal. The most fundamental of these is probably the London-Dublin relationship, as established in the Anglo-Irish agreement of 1985. While this has had many ups and downs, its strength is that the two governments have come to see their interests as being almost identical, in that both of them prize the goal of stability over everything else, including any territorial ambitions.
The next relationship is what used to be known as Hume-Adams, which began as a dialogue between the leaders of the SDLP and Sinn Fein and has since matured into the concept of political inclusivity. Just about every Irish nationalist now subscribes to this theory, which in practical terms means they believe there can be no omission of Sinn Fein from any deal.
Back when Albert Reynolds was Taoiseach the relationship between Dublin, the SDLP and the republicans used to be characterised as a "pan-nationalist front." Any resemblance to such a front, pretty much fell apart when John Bruton replaced Reynolds.
But now, with Bertie Ahern as Taoiseach, a new understanding appears to have been reached among the three nationalist elements, and in negotiating terms they all seem to be in the same neck of the woods. Although this fact is not being trumpeted abroad, it will greatly strengthen Gerry Adams's hand in dealing with dissent in the republican ranks.
The first IRA ceasefire was sold to republican hardliners on the argument that they could achieve more politically, in cooperation with Hume, Reynolds and indeed Clinton, than they could through violence. The creation of a renewed sense of nationalist unity would greatly increase the chances of the present ceasefire holding.
If there is a deal Adams and company will sell it as an interim arrangement, a staging-post on the road to eventual Irish unity. Although there are some republican dissidents who are doubtless at this moment pondering how to get bombs into England, the chances are that the vast majority of republican activists and voters will approve of this.
Adams has shown himself to be an astute assessor of the republican grassroots, as is illustrated in the achievement of pushing the Sinn Fein vote up over the years from ten to 17 per cent. He has been a moderniser but also a highly cautious one: if and when he does go for a deal, he will be confident of its saleability.
In one sense he can afford to move further then David Trimble, for most of the republican grassroots are, like the Sinn Fein leadership itself, highly pragmatic. The Unionist grassroots are much less so, being more worried and more divided and thus more problematical for everyone.
Because of this, the position within Unionism is much less certain. Unionist leaders who have shown pragmatism and flexibility have very often wound up as scalps swinging on the Rev Ian Paisley's belt, and Trimble has no desire to join them.
The next key relationship in this series is therefore that between Trimble and Tony Blair. The prime minister has given the Unionist leader unprecedented access in recent months, and been supportive towards him in very many ways. Tony Blair privately acknowledges the constant threat to Trimble from Paisley and other critics within his own party.
"Giving comfort to the Ulster Unionists is vital," the PM told a private gathering of Irish-American politics two months ago. The trick, in the final four days of negotiation ahead, will be to continue with that comfort while nudging Trimble towards the nationalist position.
This will not be easy since some Unionist negotiators, including possibly Trimble himself, have clung to the idea of finding an agreement without Sinn Fein. Some of them thought, and indeed actually hoped, that the IRA ceasefire would have broken down by now; they believed in the mirage of Dublin and the SDLP splitting off from Sinn Fein.
Because of this the psychology of deal-making has not permeated the Unionist party as thoroughly as it has most of the other parties involved. It is this which now makes the Trimble-Blair relationship so vital and so delicate: the PM must judge just how much pressure the Unionist market in general, and Trimble in particular, will bear.
The things that might still go wrong are legion. There could be violence on the outside, or splits on the inside; there will certainly be a great deal of brinkmanship before Thursday's deadline. But as against that there is the feeling that last week a little click was faintly but distinctly heard: that the sense of momentum which was missing for so long has now arrived, and may carry this phase of the enterprise to success.Reuse content