What team(s)? I'm assuming if you're starting with a big win streak then you're one of the top teams in the league. If so, its probably just a case of over-confidence and your players not working hard enough. And of course morale will drop if you lose a game you're favourites to win.
The prime example being the Chelsea team of late (that is, Pre-Di Matteo). They are all superstars. They know they are superstars and forget the importance of hard work. They then start to struggle and as a result, morale drops and the problem spirals.
The key is maintaining a balance of keeping confidence high without your players starting to think that they're gods.
That being said, this is FM and there could be a whole host of other reasons behind it. Other teams figuring out your tactics, players becoming jaded, the 'honeymoon period' wearing off (which I'm not 100% exists in FM but from what I've seen I'd argue that it does), etc
EDIT: One thing I think you can rule out, though, is that it's a bug. Yes there are bugs, as you would expect with such a detailed game, but they are usually minor issues, not things as fundamental as being able to maintain a successful run.
