Compound interest gains most of its power when large amounts have been saved. So if you don't make much money, compound interest simply won't make you rich, you won't be able to save enough (though you can still have a decent retirement). If you make a lot, it doesn't matter as much anyway. If you're middle class and willing to save half your income, then it might make you rich, but that is a painful 30-40 years. Explore the graphs and savings calculator here for examples of what you would need to do to have a million by 60.

Mostly that life is too short and interest too low for that to really happen before you die.