For those of you that know much of anything about 3d printers you know the quite obvious answer to this, but for those who do not know much about 3d printers this is a serious question that I have been asked on multiple occasions. The simple answer to this question is.. well sort of. There are many types of 3d printers out there and many custom ones being built nearly every day and many of these printers use 3d printed parts in their build. The RepRap community, which is a community behind making 3d printers accessible to everyone and be able to replicate them as easily as possible has some awesome designs that take advantage of lots of 3d printed parts.
Of course, there are parts of a 3d printer that cannot be printed for obvious reasons and that is for one the electronics. You will need a board to connect all of your hardware together and control the movement of your 3d printer. Also, the hotend and motors are things that will need to be purchased. On top of that you will likely need some additional hardware that will have to be purchased no doubt.
This leads back to my initial point of sort of being able to 3d print a 3d printer. Some of the kits designed in the reprap community use a majority of 3d printed parts to connect all of your other hardware together. This is fantastic for a variety of reasons. Firstly, filament is fairly cheap so if you have a 3d printer and want to build another, aside from time of printing, the actual parts will not cost you very much as most builds do not use more than a 1kg spool or so of plastic. It also allows for the ability to repair a part on a 3d printer very easily along with make additional modifications and upgrades to improve the print quality that can then be shared back with the community.
I got to experience this just recently, as one of the main parts on my personal prusa machines (the x-idler) snapped off. Luckily the manufacturer provided a list of all the plastic parts that had been used to build this machine to be easily downloaded. Before heading to bed the other night I started the print for the replacement part and when I awoke like magic there was the part I needed. It is quite an incredible thing, the ability to upgrade and replace parts of a 3d printer with a 3d printer. One that is definitely unique to this hobby.
There are some very creative people out there that are finding more and more ways to use plastic components in 3d printer builds to make it even cheaper and more accessible to those with limited hardware.