Welcome to the forum. I wouldn't worry too much, since we are talking about music. If we knew what country you were in it would be a help, because each place is different. I can say this though, with music in every case I have seen taken to court so far, they only went after uploaders. In fact, they went after uploaders that were sharing thousands of files. I doubt that the amount you have obtained is enough to trigger any litigation.
But to answer your question. Incomplete or not, if they show up in your shared folder they could be making you a target. Some clients, like BitTorrent will still make use of incomplete files for re-distribution. Therefore, always include your software when posting -it helps to get an accurate answer.
If you are concerned about litigation, then I would stop downloading of course. But also, you should empty your shared file folder, or at least transfer these files to another folder that is not visible by others on the Internet. Since this is how the files are distributed/uploaded from your machine and this is how offenders are picked out by copyright groups.
Having said that, do not expect such leniency with movies downloaded from the net. They don't care if it's uploaded or downloaded, they have exhibited a zero tolerance for abuse of copyright with file sharing. Even one file can get you in trouble if you live in the United States.
Thanks Taxman for moving this post for me, it is more appropriate here.