It doesn't seem efficient. Open-ended data are a burden themselves. So adding extra noise by allowing for some participants respond in an open ended question, and others not, will only lead to your closed-ended question miss data.
You can add some space for participants to ADD open-ended information, but they should all respond to a uniform set of closed-ended questions.
Nevertheless, open-ended questions and closed-ended questions can both coexist in the same questionnaire.
Of course, little can be said without more information on your research project.
I don't think it will be valid to say the open ended questions are giving the same findings as closed ended. However, a mix study or it is also called Triangulation can be used, where both types of questions are included. Usually this approach depend on both quantitative and qualitative research
Like the others, I am not aware of anyone who has done this, and I would not recommend it. The point of a survey is to get equivalent data from each respondent so that the resulting variables can be used consistently throughout one's analysis. Creating part of the data in a variable from open-ended responses and part of the data from closed ended responses would not only defeat this goal, it would be very difficult to do. In particular, people would undoubtedly mention things in the open-ended data that had no equivalent code in the close-ended data.
I would echo the answers provided by Professor Muayyad Ahmad and David L Morgan. Providing an option for both open- and close-ended questions would get a little messy.
However if you are brave you can follow each of your questions with a text box entitled "Additional Comments". You see this in online satisfaction surveys (for the purposes of marketing) all the time however I would think there is limited research value.
But to answer your question "Is there research on what type of subjects / respondents opt for open-ended questions?"... not that I am aware of. Think about this a little though. What type of person is mostly likely to opt for the open-ended format? A researcher :) or someone who has a vested interest in making sure their opinion is heard. Again, in the world of satisfaction surveys the people most likely to give you a response are those who have experienced excellent service and those who have experienced extremely poor service. These individuals have a motive for completing the survey. Depending on your sample you will most likely find there are people who will not provide information in the "Additional Comments" section while there are people who will write you a book. Those who provide a vast amount of information will have a motive or personal interest for doing so. Of course this will vary depending on the topic being surveyed. Different people will be triggered in different situations.