The exact year(s) that the war took place have never been stated, though it can be concluded that the Mushroom Bomb was dropped sometime after, at the earliest, 1985, as in "Everything Stays", Two Bread Tom, a surviving human, is depicted singing the theme song to the sitcom Mr. Belvedere, which premiered that year; in "Simon & Marcy" the Ice King is depicted singing the theme song and making reference to the sitcom Cheers, which premiered in 1982, further supporting the 1980s timeframe of the war.

The 80's?! I never thought about that! Wow!