I've encountered both. On the thugs bodies you will find a note on who hired them.
On the assassins bodies you just find a letter that some ritual was done and it's signed.
Is this some part of a quest? The same person signed both assassins notes.

You, friend, are being targeted by the Dark Brotherhood.

Watch your back.

Someone from the Dark Brotherhood tried to assassinate me yesterday and he had a note but didn't say who wanted the contract done.

If you steal something, even if you are not caught stealing, thugs will be sent after you. It was done to me.

I'm not sure about the Dark Brotherhood but if you assaulted someone and then ended up killing them you might get a Dark Brotherhood contract put out on you.

In any case, neither one is part of the main quest.

I was attacked on the bridge outside of Whiterun by an assassin Khajiit that jumped out of the bushes. Turned out the Dark Brotherhood wants me dead, and I can't even recall doing anything that could have provoked anything at all by the time that happned.