Yes, it really does. Other than the cost of the med itself, I can't find any downside in the story.
I use Betaseron and don't really have any side effects other than site reactions.
But I've read about the breathing/chest pain episodes that some people get, on occasion, from Copaxone.
This article seems to indicate that even though they increase the dose, that particular side effect is not worse...which I'm sure is good news to a Copaxone user.