You can save yourself rich. Given enough time you can accumulate a million dollars or more by savings. If you make an investment of $10,000 at 12% interest it will double every 6 years. At 10% it will double every 10.2 years, and at 6% it will double every 12 years. I used 12% for my example as that is the historic long term stock market return. That $10,000 will grow to $640,000 in 40 years exclusive of income taxes and investment fees. So find a 12% return in a tax free investment and you are good to go, well at least when you are forty years older.
I do not save money in the traditional sense of having a savings account for retirement or some other savings account type. Yes, you can save yourself rich. If that is your choice, my only suggestion is, start early, and look for any method that will increase those savings rates.
Instead of having savings (I was never very good at saving), I have developed enough monthly cash flow to do those things my family truly wants to do without worrying much about where the money will come from. Besides if I had $3,000 laying around in a savings account I would buy a rent house with it. My brother-in-law has three quarters of a million dollars in his company savings account accumulated over thirty years from monthly payroll deductions and his employer's stock purchase program.
One day some years ago he asked me how I lived better than he did when he knew he made more money than I did? I told him that we spent all of our paychecks and we have no appreciable savings while he saved some of his paycheck every month. My brother in law uses his money (paychecks) to live on and is saving for retirement, which takes part of his paycheck each month. I spend my paychecks plus some of the positive cash flow from my investments giving me a higher standard of living while I get slowly rich as my tenants pay down my mortgages.
My brother in law may well save himself rich with the help of his employer's stock plan. I let my renters and businesses make me rich. Same goal - different methods My brother in law has his money in a company sponsored savings account earning low interest rates.
I have my money working for me in investments and businesses. My money earns much higher returns than my brother in laws savings accounts, and my money provides me with monthly cash flow that allows me to do what I want, when I want. That difference in concepts can make a big difference in your standard of living. In all my conversations with wealthy people savings is not discussed.