Can people invent new colours that do not exist?

No, color is a perception and the human eye/brain sees the colors it sees. Also all the electromagnetic waves detected by the eye when it perceives colors are already there (and many others that our eyes are not built to detect, but insects' and birds' eyes can detect).

New pigments or dyes that better reproduce a color without fading can be invented though. Purple dye was once so rare and expensive that only kings and emperors could afford it, but when "coal tar" dyes were invented in the 1800s everybody could suddenly afford purple and you suddenly saw it everywhere instead of only on the king's robes.