From the way your question is phrased, it sounds like you're generically asking if you should verify something executed correctly. The answer is yes.
If it's something simple like a system call, then you would return the runtime's exit code
This is assuming whatever you executed itself returns a proper error. If I run a Bash script from Linux and it returns 2, your Java code needs to know what that means and what to do.
In Java the big things to catch are Exceptions (and NOT just catching them as generic exceptions). ArithmeticException (dividing by 0), ArrayOutOfBounds (accessing part of an array that isn't defined), NullPointer, etc. Whenever you run into unexpected or risky behavior, it's always safer to catch yourself.
The extent to which you verify as you go varies on your development environment.