Unlike a lathe, that can bootstrap itself. A 3D printer needs electronic components for its control routines. Even if we excluded those you would need something in heat resistant in the nozzle which obviously your print can not make. Or in the case of laser sintering mirrors and lenses. While you could use some trickkery around this with a lot of work and cooling apparatus you would still end up with a way inferior machine.
Similarily you would have hard time manufacturing the straight supports for the mechanisms as well as the precision screws and ballbearings. Altough a sintering machine could do somewhat good fascimiles. Not to mention the fact that these are usually bigger components than the printer can do.
But yes given a long time and a lot of manual work you could design and print one today altough it would be pretty poor, and bad value for money. would it become better? Well essentially a 3D printer is a CNC machine so it can do most of the tasks required (3D printer is a bad description for a diverse family of devices that arbitrarily exclude some and include others). Bearings and screws would be a problem still...
Would it become better? Yes. One will eventually see printers that can print the electrical conduits maybe resistors and coils, but integrated circuits will be outside scope for a while yet. But evetually yes you could build one that could replicate itself.