<p>When 3D-printing on an 20*20cm, I've heard that the quality of the printings get worse if I fill out the board... Is it true? Should I keep it to small amounts at the time or doesn't it matter?</p>

No, that's not (entirely) true. There might be some loss of quality if you print multiple objects at once, because when the printhead "hops" from one object to another it might leave a mark or ooze out some material. Also, a large number of retractions in a short period of time might lead to inconsistent extrusion.
However, none of this is particular to "filling out the board" as it happens even if you print only two objects at a time (or even when you're printing only one object with multiple islands).
It all depends on your printer (and in particular how well it handles retractions). If you're willing to do a small amount of cleanup afterwards (to remove the strings and blobs) then printing multiple objects at a time is completely viable.