For me Schwinn represents both good and bad. I come from a different perspective than most people here-- I focus on the old heavy weights, including Schwinn products. The balloon tire Schwinns from the 1930s-1950s were quality machines. They represented a sort of "Cadillac" essence: cushy American luxury that turned heads. The wealthy kids rode new Schwinns. Those that could afford them would ride second hand used ones (my grandpa got his in 1939 or so that way, and it was already 3 years old when he got it). Many today also forget that adults often-relied on these machines as transportation if they couldn't afford a car (as many couldn't in the 1930s). If there was any stigma attached, it was that Schwinns weren't as affordable as they could be. But interestingly they didn't fully cash in: these bicycles are rarely Schwinn-branded. You'll see Admiral, Henderson, Excelsior and a ton of other oddball names. They're all Schwinns from the same factory, but often they don't tell you in as many words.
After WWII they continued that "top of the line" position, but now were quite proud to display their own logo, especially in the heavyweights. But part of the problem was the automobile. In the 1920s and 1930s adults actually rode bicycles, because these people could not afford cars. This included the heavy weights. The tankless "roadster" double parallel curve bar frames are examples of bicycles made to be more "mature" and affordable than the loaded-down deluxe ones the kids rode. But by the 1950s most people could afford some kind of car or another, and the depression was well over. The bicycle got relegated to a kids' toy and Schwinn never really broke free of that. They did make some excellent lightweights, but the market just wasn't there in the US, and Schwinn remained a domestic brand really. The matter, as I see it, was that the european brands were able to maintain their reputation as builders of quality adult performance bicycles (no not Viagra performance, but road performance), because their markets supported it. In America in the 1950s, the bicycle was really more the analog to a bb gun.
Schwinn was able to remake itself on some level with some decent lightweights in the 70s and 80s, but they still had that "old school" heavy 1950s bike essence about them. I think there might well have been a stigma of being heavy and backwards there. I see the road bike boom as a sort of countercultural movement against the more traditional 1950s. As part of that counter movement, I tend to think Schwinn was associated with traditional American bicycles-- the 1930s-50s. By contrast, the imports were new and exotic and may have had an element of cachet-rebellion in them against the old guard in the US.
In modern times they've really dropped the ball. They do make some nice bicycles, but the ultimate bastardization comes when they do this "Schwinn approved" thing and will allow cheap license-built Chinese bicycles from Wal-Mart to hit the street as "Schwinn". Maybe the Caillac analogy continues to apply-- some of you probably remember that phase in the early 80s when Cadillac tried to compete with the imports by making "small economy Cadillacs". They were basically re-badged low-end Chevys. Well they took a hit, like Schwinn will with this Wal-Mart business.
My nephews are now at the age when I discovered vintage bicycles, specifically prewar Schwinn (in the form of a 1936 Henderson roadster). But they know Schwinn as the "bikes at Wal-Mart". In contrast, my grandpa, dad and I all think of Schwinn as a highly respectable maker of old bicycles. So, while I think Schwinn did have a somewhat backwards stigma a couple of decades ago, I think they have a far worse problem now. They're going the route Dayton/Huffman went in becoming the much-loathed "Huffy". If Schwinn had a problem in the 1970s and 80s, they have a catastrophe on the way now. (sorry for the long post, but it just seems a long-running topic)