Usually a person who saves money has a degree of self-discipline and is lacking extensive financial smarts.
Of course, the self-discipline is necessary and commendable - but self-discipline alone won't make you rich, you have to use your brain as well.
The problem with saving money is largely the wrong attitude towards money. We like to think that we own money, that our money is value we possess. That is not quite true; the value of money comes from the flow of money and not the possession of money.
In other words, if you don't use money but just pile it up - it's the same as you had no money at all.
So if you want to be smart about saving money, invest it! Create money flows which will allow it to retain (and hopefully increase) value.
Money comes and goes; make sure it comes and goes the way you want it to.