Question: By default, in Windows, when you right-click a window title bar and select "move", it will allow you to move the window using the arrow keys. The thing is, Windows are moved several pixels each time you press an arrow key. 

So, can this be set to move the window 1 pixel each time?

Also, are there any 3rd party programs that will allow me to do this?

Answer: 
Hold down the `Ctrl` key while pressing the arrows for single-pixel control. You can press and release during the move (or resize) as needed to switch between jumping and single pixel.
Thanks a lot, great answer, though more specifically, for English language Windows 7 shell layout on Window's windows that are **NOT** "Maximized"/"in full screen mode":
`ALT`+`SPACE`->`M`->`CTRL`+`ARROW KEYS`->`ENTER`

Example: to move a window some pixels to the right: Press `ALT` and `SPACE` together. Press `M`. Press `CTRL` and `→` the number of pixels you want to move. Press `ENTER` to lock the window.

Tip: press `ESC` instead of `ENTER` if you want to reset the window you're moving.