Is downloading a copyrighted game illegal

Downloading a Copyrighted game is illegal in most countries. There are a few countries in the world that do not support International Copyright Laws nor do they have laws that deny you to download Copyrighted digital material.
As a general rule of thumb - yes.

If you own a copy of the game, you may have an additional copy for backup purposes if your country laws allow it.

US & Canada - yes
UK - yes
Europe - depends, yes
Australia - no (that is where I am from)