The Cincinnati Bengals won their conference championships in 1981 against the Chargers, but lost in the Super Bowl game that year. The Bengals also won their championship in 1988, but again lost in the Super Bowl. They have not appeared in a conference championship game since 1988.
1998 they lost to the Bengals the score was Bengals 34 packers 25
1982 and 1989
January 22 , 1989 they lost Superbowl 23 to the 49ers 20 to 16.
The catch was actually made in the January 10th, 1982 NFC Championship game against the Dallas Cowboys, not the super bowl. The Niners went on to beat the Cincinatti Bengals in the Super Bowl.