Actually, what would be best is putting money into a savings account as you get it. Once you have enough money to afford a system, spend it all at once.
As a rule, the price of computer components goes down over time, not up. Any money you spent at the beginning of your upgrade odyssey bought you a less powerful part than it would've 3-6 months later. Getting less value for your money is an important reason not to buy slowly over time, but it's not the only reason.
The other important reason is that you have no way of testing individual parts until you have all the pieces. By the time you have all the pieces you'll be way outside the retailer return window for most of them. They'll still be under warranty, but dealing with the return department of a manufacturer is generally more of a hassle than the return department of a retailer. (Not to mention that if more than one part is bad you have to deal with multiple reps instead of just one.)