In 1973, the NHTSA regulation FMVSS 216 went into effect for passenger vehicles. The rule stated that cars' roofs had to withstand a force of 1.5 times the weight of the vehicle with no more than 5 inches intrusion. This was meant to protect occupants in rollover accidents. The regulation of course would've made convertibles illegal, but the NHTSA chose to exempt them. The regulators thought this would be a moot point because droptops were a rapidly disappearing breed as drivers preferred air conditioning and sunroofs over the annoyance, weight, and complexity of a removable roof. Indeed, in 1976, the last convertible out of Detroit, the Cadillac Eldorado, went out of production. But then in 1983, Chrysler cut the roof off the Lebaron and Detroit was back in the business.
So what if the regulators decided that droptop fun wasn't worth the safety risks? This of course wouldn't just get rid of American made convertibles, but also all the import convertibles. Also, the loss of the US market would probably cause most automakers to stop selling them altogether.