Yes. You want to create a scroll region or "subwindow" on the console.
This is, unfortunately, not all that simple, especially if you plan for your program to work cross-platform.
What you should do is look into NCurses (*nixen) / PDCurses (Windows). This is the simplest way to do it.
If you have already invested a lot into making your program work with the standard terminal streams, that is an easy fix. Write yourself a custom streambuf that outputs through the Curses functions, and at the beginning of your program make sure to swap out the standard streambuf with your custom one on the cout and cerr objects. (Make sure to do the same for cin if you want to be complete.)
Alternately, if you know that your application is going to be used in a GUI system, you can just set the window title. This is a pretty easy thing to do cross-platform on Linux, and very easy on Windows.