Michel Barnier, the EU negotiator, used the d-words, “deadlock” and “disturbing”. He might as well have just said “doom”. He also said that, although he hoped to make progress, there was “no question of making concessions” on citizenship rights, the Irish peace or the financial settlement.

That meant that David Davis, the Brexit Secretary, was reduced to pleading with the leaders of the 27 EU countries to loosen Barnier’s negotiating brief to allow him to make progress.

Of course, both sides could be posturing to try to increase their negotiating leverage. The cheerleaders for Brexit cannot decide whether they think the EU will come round in the end, or whether we ought to just quit now without bothering with the details.

It was notable that Theresa May, in her Commons statement on Monday said she expected the negotiations to be concluded quite close to the deadline, which is sometime next autumn. But it was also notable that, after a mini-Brexiteer rebellion in the Cabinet on Tuesday led by Michael Gove, she also emphasised the Government’s preparations for a no-deal Brexit when she spoke in the Commons yesterday.

And there were good reasons for thinking that the Brexit talks could fail to reach agreement, even before Barnier’s doom-mongering this morning.

Barnier: The Brexit agreement will not be based on concessions

We were reminded of one of them yesterday when Arlene Foster, leader of the Democratic Unionist Party, said she would not accept a customs border “around the island of Ireland”. That would look too much like a step towards a united Ireland. But the problem is that the DUP has also said it won’t accept a border with physical barriers between Northern Ireland and the Republic. If the UK leaves the EU it has to be one or the other, despite ingenious proposals for a “virtual” border policed by digital tracking.

As difficult as that is, it is only one detail of a Brexit negotiation that piles implausibility on impossibility. Theresa May repeated on Monday that she intends to negotiate the “end state” – that is, our long-term relationship with the EU – by March 2019, and that the transition period will purely be for the purposes of implementing this agreement. I have not come across anyone with any knowledge of trade negotiations who thinks this is possible.

The best the Prime Minister could hope for would be a “heads of agreement” document, which would then need at least two years’ further negotiation to turn into an EU-UK treaty.

And that is before she considers the complications of managing Parliament. Yesterday, Jeremy Corbyn’s spokesperson confirmed that Labour MPs would vote against a no-deal Brexit. It is not clear what the alternative to leaving without a deal would be – but this does raise the prospect of the Government being forced to ask for an extension of the Article 50 deadline. And it is even less clear whether the other EU countries would allow such a thing. It all depends on whether they want to make things difficult for us, in the hope that we change our mind about leaving.