In theory, I imagine you can, but there are some practical considerations that might need some thought:
If you have a desktop printer with multiple extruders, you could probably print with both one conductive and one structural filament, and thereby build circuits in 3D.
One concern would be the low melting points of most 3D printed filaments, since one would have to limit the heat generated by the mounted components and connections so that the structure of the "board" would not be melted.
Mounting components to the board would also differ from a normal PCB, since you would have to connect the components with the conductive filament without melting the structure of the board. In other words, you probably would have to use conductive filament as "solder", and melt components into place.
Whether you could use typical tools used for mounting SMD components with conductive filament as solder is beyond my knowledge.