It looks like
FindBugs can warn you if an exception is "dropped": see
this. There might be other static code analysis tools that would warn you about the same.
It probably won't warn you if it is not properly logged, though. That should not be so much of a problem,
you should log exceptions at the topmost level of your program, and probably install an uncaught exception handler to log those that might otherwise be missed. (You might need to setup a Swing-specific uncaught exception handler if you're using Swing.) These places should be few and thus relatively easy to review.
If you're logging and dropping an exception at a lot of places in code, this won't help you. Such a design would look very suspicious to me, though.