Daniel,
We sometimes have a similar problem when students don't wait for the submission to complete but use the back or the bread crumbs to go back to class and work. They need to make sure they wait for the message that the submission has been successful. They will also get an email if they have successfully uploaded their assignment.
We have also found that if we are restoring or have reset a course with Turnitin assignments we have to reset the assignment or they will see the past semesters students and submissions as well.
Paula