Not given what I've heard abou the American system, no.
There is really, as far as I know, only one serious problem with Swedish health care and that's the waiting lists. People sometimes wait for years with broken hips and stuff like that because the queue for an operation is so long.
The waiting lists depends slightly on where you live though. In Stockholm and other crowded places I've heard of people waiting for that long, but around here the list is quite short. My sister has had quite a lot of troubles with her legs and the longest she's ever had to wait for surgery was two months, and that was something very minor. It's still a serious issue they're so long some places though.
I also doubt the trouble with the waiting lists would be eliminated if we didn't have public healthcare considering the fact that the waitinglists when it comes to dental is even longer, and dental is not public. People can wait two years just to get braces.