Late 80s, early 90s. roughly that zone where the NFC won 12 or 13 in a row, and most of those were Dallas, The Giants or the Niners. It was a boring era to be an AFC fan. Not a terribly exciting era to be a fan of an NFC Central team either, until Greenbay got the last win of that run.
This wasn't really a function of NFC dominance, in my mind. The real issue was two-fold:
1. In the 1980s, the AFC went through several years where the AFC team that would match up best against the dominant NFC teams would lose in the playoffs. The Broncos of Super Bowls 21 and 22, for example, were not great teams from top to bottom but had some solid players who came up big in the playoffs. I still say to this day that we may have seen two of the best Super Bowls of all time if the Cleveland Browns had won those two AFC championship games. Those Browns were built like some of the best NFC teams of that era.
2. In the 1990s, the "run and shoot" took hold in the AFC -- with disastrous results outside the conference. Maybe they were just 10-15 years ahead of their time, but that type of offensive game plan didn't work well in a tough game against an opponent with a great ground game. The Bills demonstrated that in spades, in both victory and defeat. The AFC wild card game against the Oilers in 1993 exposed all of the flaws of this offense; Warren Moon threw for over 200 yards and 4 TDs in the first half alone, but Houston blew a 32-point lead in the second half even while playing against an injury-depleted Buffalo team. The "run & shoot" offense of Houston was exposed badly, as it was completely ill-suited to mount time-consuming drives with a running game. That may have been the first NFL team in my lifetime to go through a season without a tight end on the roster. They just lined up with four wide receivers and threw the ball downfield all day long.
The NFC won every super bowl between 1985 & ‘97. I liked some later super bowls more. When I was a kid, I lived near Las Angeles and Denver, so my two favorite teams are the Rams and Broncos. The Broncos won super bowls in 1998 and ‘99. The Rams won the super bowl in 2000, and they lost the super bowl in 2002.