Most of Holden Caulfield's assertions throughout his narrative in The Catcher in the Rye are either evasive, defensive, or offensive. This is partly the reason why he hints at "getting sick" prior to being admitted into a rest home. We will find that it is in his way of conveying facts, and in the rationale that he needs to justify himself, that he hints this information to the reader.
First, let's examine Holden's way of conveying facts. In some instances, he is able to break down his thoughts into examples and get to the point of things. Overall, Holden combines naivete, angst, and evasion of facts, to create a method of communication in which he does not have to be held responsible for his opinions, in general. He tells us facts, but he chooses whether to explain them further, or not.
However, in chapter 26, he seems to recoil from having said "too much". Since he is now a bit ashamed of telling his story, he would naturally have to admit that something else was wrong prior to being admitted at the rest home: he needs to provide justification.
By stating that he "gets sick", he is telling us that he is in the rest home NOT JUST because he needs psychological help, but because a combination of factors lead him there. If he doesn't hint at the fact that he gets sick prior to being there, then his entire testimony would have rendered him a complete "mental case". Holden's ego is too fragile to allow us to come to that conclusion.
Therefore, the combination of his narrative style, and his need to justify his stay at the rest home, are the reasons why Holden finds it necessary that we understand that he is not necessarily a "nutcase", but that he is also a victim of illness. All those factors make up his "real" reason why he ends up where he is. Although he seems to be making progress at maturing, Holden is obviously not completely ready to admit to some things. Not "just" yet.