Can a 3D printer make a 3D printer?
Yes, you can print most of the parts (electronics, linear guide rails, ball bearings and nuts and bolts, etc cannot be printed). Actually this was exactly the purpose of RepRap.org:


  RepRap is humanity's first general-purpose self-replicating
  manufacturing machine.


and:


  Since many parts of RepRap are made from plastic and RepRap prints
  those parts, RepRap self-replicates by making a kit of itself - a kit
  that anyone can assemble given time and materials.


There have been attempts in the past to even replicate the frames of printers (e.g. Dollo 3D or Snappy, but such designs are not very successful, printed frames are more flexible than metal frames.



I have built 2 custom printers myself using other printers to print parts and printed all printer parts for several others. It is possible to print your own linear bearings from POM, I prefer these over the noisy metal bearings.