And, in days such as these, politicians who have a sense of duty to their country, who have people's best interests at heart and who are capable of distinguishing reality from fantasy know that the normal rules do not apply.
Diplomats said the two sides were moving towards a separate legal statement - a "parallel declaration" or "interpretative instrument" - on the backstop.
The stalemate has raised fears of Britain leaving without a divorce deal in place at the end of March, an outcome that could cause severe economic disruption.
A second diplomat, briefed on the May-Juncker talks, confirmed the European Union would only signal this was the direction of travel, rather than offering a precise text, before May faces another vote in parliament.
However, Mr Juncker had a more downbeat assessment of the situation when he stated the next day that he was "not very optimistic" that a no-deal Brexit could be avoided.
May will try again on Wednesday, but on this trip she will not even meet President of the European Council Donald Tusk, who represents the leaders of the EU states.
"We recognise that a no-deal Brexit would be a very bad outcome for the United Kingdom and we are doing everything we can to avoid that", he said.
"But that will depend on the progress that's made over the next few days. That is why we are determined to get no deal off the table".
No.10 said talks with Mr Juncker had been "constructive".
Percy, who voted to leave the European Union, told BBC Radio 4's Today programme "Some of my colleagues have got to recognise that the game they have thus far been playing with regards to this whole process is not going to end well for them and could potentially end with the delaying of - perhaps even no Brexit - which some of us have spent a lot of our parliamentary and political careers campaigning for".
Meanwhile, the European Union expects May to be forced to request a three-month delay to Brexit, two European Union officials said.
Alternative arrangements could replace the backstop - but only in the future.
Despite officials in both Brussels and London playing down suggestions that an EU-Arab summit this weekend could become a Brexit "deal in the desert" meeting, Hammond added May's attendance was an "opportunity to talk to some of the European counterparts there". The defections involve only a small fraction of the 650 lawmakers in the House of Commons, but mark the biggest shake-up in decades for Britain's political parties.
The Daily Telegraph reported that Mr Cox is drawing up plans for a unilateral exit mechanism from the backstop with a 12-week notice period.
After the Brussels talks, the PM said: "I have underlined the need for us to see legally binding changes to the backstop that ensure that it can not be indefinite".
"We've agreed that work to find a solution will continue at pace", May told broadcasters after their meeting.
"Whilst we fully expect some changes to the backstop arrangements to be made by ministers in Brussels this week, there remains a chance that these will not satisfy some colleagues".