Suppose I am improving a bound in a problem in "some cases" by using the extra information available in that problem which was unused in the earlier bounds for that problem. However, in general, I am not able to show that my bound is better than the earlier bound. Can this result be considered to be a reasonable result?
The result might or might not be reasonable. The following would speak for it being interesting:
The special case is interesting. A good argument for this is that others have looked into it or that it has applications or other relevance in other fields of science or mathematics.
The method of proof is new.
But, in general, the only way to know is to write it down, talk about it and submit it to a journal and see what the response is.
When writing the result down, you will want to emphasize the relevance of the result and refer to similar results, applications and the new features in the proof.