Framton Nuttel is essentially alone, nervous, and depends on others. We learn this when he says that he is in the country in search of or a cure for his nerves. He explains twice in the story how much he needs his cure, first to Vera, and then in detail to Mrs. Sappleton.
The doctors agree in ordering me complete rest, an absence of mental excitement, and avoidance of anything in the nature of violent physical exercise
Already, this characterizes him indirectly as a frail and weak man whose nerves could betray his state of mind at any moment. After all, he is just then trying to get the cure--he is not cured yet.
A secondary clue is the way in which he lands in Ms. Sappleton's house. It is through his sister, and not by his own doing, that he is able to get there in the first place. Mrs. Sappleton is a lady who had made the acquaintance of Framton's sister a while back. It is clear from the sister's words that she is not entirely comfortable sending Nuttel there, but she also seems interested in getting rid of him, or else in him getting rid of his issue. She is not very supportive, either:
I know how it will be [...]you will bury yourself down there and not speak to a living soul, and your nerves will be worse than ever from moping. 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...
Evidence that Framton is impressionable is found in part 2, where you can already tell that Framton is engaged in Vera's story because he starts asking her questions.
"It is quite warm for the time of the year," said Framton; "but has that window got anything to do with the tragedy?"
This reinforces the assumption that Frampton is a vulnerable man whose receptive and submissive personality renders him more prone to fall for the stories of others. Even though nervous conditions are not indicative of a frail character, they do place those who suffer from them it in a position where others who are stronger could take advantage. This is exactly what happened between Vera and Nuttel.