The system doesn't just check if you can execute the command, there are many commands you can execute. The OS checks if whether you have the rights to make the change you're trying to make. If you do not, then the program you're trying to run is sent an error after it tries to do the operation.
For example, if you want to run the chmod command on someone else's files and you have no group permissions and the file isn't +w for the "a", so something like rw-r--r--, then there is simply no way for you to edit or modify the permissions with the command if it's "jim"'s file and you're "joe". The operating system is preventing the action, it's not the program itself. The program will only be reporting the message back to you. In fact, the command probably actually try to do what you asked... even though it cannot.
Basically, Linux has a file security infrastructure like this:
Root > Owner > Group > All Others
So, someone who is "more left" in that list has rights to change the permissions or data of anyone to the right of them where the "All Others" have none.
But, then you have a more complex issue... People can run a program as "setuid", "setgid", or whatever. That means the program runs as the user/group combination that is attached to the file. It's sometimes used to give someone access to a program. (Not recommend to use, what is recommended is giving the user adequate permissions.)
Anyway, just because you can run any program doesn't mean you can do anything with it. :D