Unlike many elite QBs, Marino started out on an already built elite team. After Greise's 1979 retirement, Shula went 8-8 in 1980 -- a rebuilding year behind the burgeoning "Killer Bs" defense. In '81, the Killer B's matured, added a few key components (Brudzinski, Lyle Blackwood) and turned the Dolphins into an 11-win squad.
Without looking it up, I believe the Dolphins were either the #1 or #2 AFC playoff seed for five straight seasons (1981-85) with two AFC titles (1982, 1984). With the Killer Bs still at the height of their abilities, Marino took over for a hurt & ineffective David Woodley during the fifth game of the 1983 season and went 7-2 as a starter in the subsequent nine games.
On those solid early 80s Dolphins teams, they actually had a credible "Thunder and Lightning" rushing attack with Tony Nathan and Andra Franklin. Marino stepped into a great situation with one of the AFC's top teams.
The Dolphins did lay some playoff eggs to be sure. To end Marino's rookie year, the Dolphins turned the ball over five times (two Marino INTs) and lost in Miami to an green Seattle team that had to travel cross-country. And then, in January 1986 with a third AFC title in four years on the line, the home-standing Dolphins turned the ball over SIX times to a good-not-great Patriots squad. Recall that the Dolphins were the Bears lone 1985 loss (albeit in Miami) and also recall the shellacking Chicago put on New England in SB XX -- a missed opportunity for Marino to be sure.
Meanwhile, starting in the 1984 off-season, a few wheels started to fall off. Defensive coordinator par excellence Bill Arnsparger left Shula's staff to coach LSU. Promising running back David Overstreet died in an auto accident shortly before 1984's training camp, robbing the Dolphins of Nathan's successor after the draft. The Dolphins traded for LB Hugh Green in 1985, taking a gamble that Green was fully recovered from a facial fracture suffered during a 1984 car accident. Green did start for the Dolphins for several seasons, but was not the force he was with the Buccaneers before the accident.
By the beginning of the 1986 season, the remains of the Killer B defense had either moved on, gotten old, or gotten hurt. The Dolphins gave up 30+ points in four of the first five games that season, including 50+ twice. While the Dolphins offense still led the NFL in scoring, the defense was 26th in points allowed (after finishing 7th in 1984 and 12th in 1985). 1986 was Marino's last season with a credible rushing thread (Lorenzo Hampton's career year - 1276 yfs, 12 TDs). But by that season, "Chuck Studley" (Arnsparger's replacement) had become dirty words in Miami.
The Dolphins wouldn't surpass 8 wins again until 1990 -- just in time for the Bills fresh new juggernaut to slap them out of the playoffs in freezing Buffalo. Buffalo would repeat the favor in the 1992 playoffs. Shula had built the Dolphins defense back to respectability by this time behind guys like John Offerdahl, Louis Oliver, Tim McKyer, and Bryan Cox. The Dolphins offense force-fed running backs like Sammie Smith and Mark Higgs, but could never sniff 4 ypc. So opponents were able to mitigate the Dolphins' passing game, especially as guys like Mark Duper and Mark Clayton started aging (yep, they were still starting).
So when you start breaking down the first decade or so of Marino's career, it's not hard to tell the story of Miami's lack of playoff success season-by-season. After that, the Dolphins went to the playoffs five more times between 1994-99 (Marino last seasons), but never got out of the divisional round. There were a few Dolphins wild card teams in there coming out of a weakening AFC East. I don't think those late 90s Dolphins teams were anywhere near as good as the early-80s Dolphins teams. Or even the 1990-92 teams.