Move running program to background in bash

I do not think it is possible to move a running program to background(while the process is still in running state).
However there is an application called GNU-Screen,using which you can run any number of console-based applications--interactive command shells, curses-based applications, text editors, etc.--within a single terminal.
Here is a nice tutorial on GNU-Screen.
Note that it does not do exactly what you asked,but serves the purpose anyway.
Example:
Here is the implementation of what you asked using GNU-Screen.

1. Run screen
   You will get details about the software.Press return.
2. Now,you will see your normal terminal.However it is a window of screen.Type any command say vlc. It will start the process within that window.
3. Now to open a new window type Ctrl+A and then C.You can run a new command here.Note that the process which you started in window 1 is still running.
4. To detach from screen press Ctrl+A and then D.You now return back to your old termianl.However any process that you started from screen would still be running.

To my best of knowledge you could run it and then append an & after it like this: myprogram &

I may be very wrong here however.

[EDIT]: Taking a closer look to this, if you run the above, the application will close when you close the terminal. If you would like the application to run in the background, then, as far as I know, the application needs to be written in such a way that supports daemon behavior.

[EDIT 2]: nohup command & appears to be closer to what you want to achieve. Nohup will allow an application to run, and will both ignore its input and redirect its output to a file called nohup.out in the current directory.