Early in the story there are several passages of explication and of dialogue which make it perfectly clear that Framton Nuttel has never met the Sappletons before his current visit. Here is one of the most obvious:

Privately he doubted more than ever whether these formal visits on a succession of total strangers would do much towards helping the nerve cure which he was supposed to be undergoing.

The Sappletons must be total strangers. Such visits must be awkward for both Framton Nuttel and the people upon he is imposing with his letters of introduction from his sister, a woman they knew only casually and haven't seen in four years.

Another obvious sign that Framton has never met the Sappletons before is contained in the following:

"Then you know practically nothing about my aunt?" pursued the self-possessed young lady.

"Only her name and address," admitted the caller. 

Vera, the self-possessed young lady, is fishing for information. She ascertains that Framton doesn't know anything about anybody in the whole region. He is the perfect victim for the practical joke she has in mind.

When Framton's sister had offered to give him the letters of introduction he was now putting to use, she had told him that some of the people, as she remembered, were "quite nice."

Framton wondered whether Mrs. Sappleton, the lady to whom he was presenting one of the letters of introduction came into the nice division.

If Framton was presenting a letter of introduction to Mrs. Sappleton, it is obvious that he has never met her before. Also, if he had met her he would not be wondering if she were "nice."

Saki takes considerable pains to establish that Framton doesn't know anybody in this region. Framton is a good example of how a skillful fiction writer will create characters to suit the needs of his plot. Vera can tell him a preposterous story about the three hunters being sucked into a bog because he knows nothing about the family. Furthermore, he is undergoing a "nerve cure," which makes it likely that he will have a violent reaction when he sees the supposedly dead hunters approaching the open window in the gathering twilight.

Speculative Analysis

Framton has evidently met other people with his letters of introduction. Is it likely that one of them might have said something about the male members of the Sappleton family which would indicate that they were alive? To forestall this question, Saki may have emphasized that Framton is only interested in talking about himself and his ailments and infirmities. When he first meets Mrs. Sappleton he tells her:

"The doctors agree in ordering me complete rest, an absence of mental excitement, and avoidance of anything in the nature of violent physical exercise," announced Framton, who laboured under the tolerably widespread delusion that total strangers and chance acquaintances are hungry for the least detail of one's ailments and infirmities, their cause and cure.

And after Framton has fled from the house and Mr. Sappleton asks his wife:

"Who was that who bolted out as we came up?"

She tells him:

"A most extraordinary man, a Mr. Nuttel," said Mrs. Sappleton; "could only talk about his illnesses, and dashed off without a word of goodbye or apology when you arrived. One would think he had seen a ghost."

In the first few paragraphs, the author tells us that the Sappleton's were total strangers and it also says that Framton's sister gave him a letter of introduction, indicating he did not know this family already.

In Saki's craftily executed frame story, not only is the main character, Framton Nuttel, taken in, but so is the reader.  Perhaps Saki's greatest story, "The Open Window" begins with the frazzled Framton Nuttel arriving at the Sappletons, whom his sister knows, with a letter of introduction which she has provided him because she worries that he will "bury" himself in his rural retreat meant to be a nerve cure and not speak to anyone.

As it turns out, Framton is held captive by the niece of Mrs. Sappleton, and she is a young lady whose talent lies in blurring the line between imagination and reality so well that poor Framton is terrified after gullibly believing the niece's tall tale about her uncle and cousin.  So, rather than working a "nerve cure" for Framton Nuttel, she has unnerved the man so much that he jumps up and flees without even saying goodbye.