Constant-time computing is used to avoid revealing information about secret values. If the machine acts differently, e.g. takes longer to respond or uses different resources to generate the response, then that says something about the relationship between a guess and the secret value.
If the processing of all failing requests is indistinguishable then that should prevent an attacker from gaining (most*) information about the secret value. An attacker gets to know if their guesses are correct or incorrect but this doesn't allow for any attacks more efficient than brute-force.
I am wondering how one would implement this through. If the program is to execute differently based on whether the guess is correct or not then this is a constant-time comparison that needs to be made first, and then you might as well just return the result of that comparison.
Here's an article that presents some common pitfalls when implementing constant time functions, and I think it brings up points that are relevant for this question.
* An attacker could potentially gain some knowledge about the secrets by eavesdropping on legitimate users and seeing the difference in processing between their successful and unsuccessful guesses. This would be a much more difficult attack to perform than if they could analyse their own requests and is probably (from my understanding) not viable in the vast majority of cases. The exception would be if legitimate users send many correct and incorrect guesses to the application.