It is not true. Success tends to breed its own set of problems. Everyone has problems. Accept the fact that everyone has problems. This will help you to move on with your life rather than sitting and feeling pity for yourself.
Every mountain has a peak and every valley has a low point.
Similarly, life has its ups and downs. No one is up all the time or down all the time. Problems do get resolved in the long term. They don't last forever. History teaches that every problem has a limited life span.