This is a pretty old article by Kerry Byrne that I always thought was fun. It's dated now (from 2008) so some of the rankings could shift, but not significantly. But I think they had a very good comment regarding the Patriots: looking at their 1960 - 2000 teams and their accomplishments, they looked extremely similar to the Cincinnati Bengals from 1968-2007.
A CHFF epic: all-time franchise rankings
The Patriots franchise from 1960 through 1988 (29 seasons) had a regular season record of 202-209-7. They also had the aforementioned 6 playoff appearances in 29 seasons. This is pretty much the definition of a mediocre team -- sometimes bad, sometimes good, but rarely upper echelon. A few good teams didn't make the playoffs: the 1964 team went 10-3-1, 1961 and 1962 had identical 9-4-1 teams miss the playoffs, the 1977 9-5 team and the 1980 10-6 teams missed the playoffs. The stinkers in that bunch were: 3-10-1 in 1967, 2-12 in 1970, 3-11 in 1972, 3-11 in 1975, 2-14 in 1981. 5 in 29 is more than we'd want, but it's not 100% terrible either when balanced out by the decent squads.
I think the perception of the Patriots as a laughingstock pre-Belichick/Brady or pre-Kraft teams is really based on the following two items:
1) The Sullivans and their clown-show, shoe-string operations as owners, and
2) The missing period outside of my analysis, the 1989-1993 teams, which were truly terrible, and included the disaster that was Victor Kiam and other scandals.
But the team itself was just meh, generally not horrible, but bottom third of the league as you put it sounds about right. Through that time period I think the Bucs and Cardinals were synonymous with a horrible franchise, but the Pats weren't that far above them.