The history of literature is full of examples of story sequences in which each individual story is credited to a distinct narrator: Chaucer and Boccaccio should leap to mind; Straparola and Basile are familiar to fairy tale readers; and Eustache Le Noble's 1697 story sequence Le Gage touché even uses parlor games and the practice of redeeming pledges to develop the frame story for its separate stories with separate narrators. In all of these, narrators may be interrupted, but the story generally remains theirs to tell.
Below, de Voyer describes something different: an informal game of collaborative fiction in which each narrator contributes in sequence to the same ongoing story--i.e. a round-robin story, interpreted as a game. He offers advice on how to make the game go well and includes assurances that it is charming, implying he has played it, but whether he has is unclear.
As his primary example, he points to a 1777 reprint of a 1673 story sequence by François Hédelin, abbé d'Aubignac, an excerpt from which is translated here. The story sequence has a frame story in which the characters, tired of their ordinary parlor game penances, arrange to play a game of collaborative fiction, and the story emerging from that game forms a large part of the book. However, the book depicts collaborative storytelling in fictional terms, so the concept of an "Interrupted Story" is demonstrated, but its execution seems to be fictionalized. The possibility remains that the account here--very similar to the fiction in its particulars--is based on a fiction inspired by parlor games.
However, another example from 1701 can be found in Charlotte-Rose de Caumont de La Force's Les Jeux d'esprit, in particular under the heading "Le Jeu du Roman" which was also the name of a game mentioned in passing in Madeleine de Scudéry's "Les Jeux, servant de préface à Mathilde" in 1667. Of course, parlor games then did often involve telling stories (telling stories that illustrate proverbs or just plain telling stories, which de Scudéry also mentions), and in telling stories, other kinds of collaboration doubtless occurred.
Interrupted Stories are one of the most pleasant amusements, provided that the last place falls to a person who has enough wit to be able to form a pleasant and singular dénouement. You will find an example of such a game in the Bibliotheque des Romans, volumes of January & March 1777: the story is entitled Les Amoureux de Quimpercorentin. He or she who begins a story of this kind must clearly establish the state of the Hero & Heroine of the Romance; make them know each other, and make them fall in love with each other, with some sort of verisimilitude; and then, let him leave them in such embarrassment as he wishes: this will make his successors pull them out of it, or throw them back more and more, until finally the last, who will have loaned the greatest attention to that which the others will have said, during the short space of time that they will have been told not to overspend, finishes the story with a dénouement such as their wit will suggest to them. When one knows well how to arrange this little game and if most of the actors are people of wit, it is charming.
Les Histoires interrompues sont un des plus agréables amusemens, pourvu que l'on fasse tomber la derniere place à une personne qui ait assez d'esprit pour pouvoir y former un dénouement agréable & singulier. Vous trouverez l'exemple d'un jeu de cette espece dans la Bibliotheque des Romans, volumes de Janvier & Mars 1777 : l'histoire est intitulée les Amoureux de Quimpercorentin. Il faut que celui ou celle qui commence une histoire de cette espece, établisse bien clairement l'état du Héros & de l'Héroïne de son Roman ; qu'il leur fasse faire connoissance, & les rende amoureux l'un de l'autre, avec quelque sorte de vraisemblance ; & ensuite, qu'il les laisse dans tel embarras qu'il voudra : ce fera à ses continuateurs à les en tirer, ou à les y rejeter de plus en plus, jusqu'à ce qu'enfin le dernier, qui aura prêté la plus grande attention à ce qu'auront dit les autres, pendant le petit espace de temps qu'on leur aura prescrit de ne pas passer, termine l'histoire par un dénouement tel que son esprit le lui suggérera. Quand on sait bien arranger ce petit jeu, & que la plupart des acteurs sont gens d'esprit, il est charmant.