What is the problem with Framton Nuttel in "The Open Window"?
The problem with Framton Nuttel in "The Open Window" is that he is presumably coming back from a nervous breakdown. As a result, he has to find a way to rest and recover fully before he can be back in the city. In synch with the social conventions of his time, Nuttel would have wanted to find an acquaintance with a place in the country to get his bucolic rest and get better.
The doctors agree in ordering me complete rest, an absence of mental excitement, and avoidance of anything in the nature of violent physical exercise
Yet, we can argue that he had another problem. The house that he was going to is the property of a Mrs. Sappleton, which is a past acquaintance of Nuttel's sister. It is traditional that Ms. Nuttel would have given references about her brother in introductory letters for him to take back to the people she once met while staying at the rectory. This, in hopes of him being able to spend time there. However, the niece of the lady of the house, 15-year old Vera, has made Nuttel more uneasy with a fake story that involved 3 dead men who may or may not come back from their deadly hunting trip at any given moment.
Nuttel is already odd, and his sister had warned him.
...you will bury yourself down there and not speak to a living soul
Moreover, he is pretty much on his own:
I shall just give you letters of introduction to all the people I know there. Some of them, as far as I can remember, were quite nice.
Therefore, his problem combines his nervous condition, his current situation, the lack of real support and Vera's imaginative story. All worked together against him, really, for he ended up way worse than before.