Can I fine-tune my screen brightness?

A somewhat overly technical answer, but you did ask "is it possible..."

Open Terminal and go to /sys/class/backlight, then into the backlight control's directory (the name varies, acpi_video0 in my laptop), then read the files max_brightness and actual_brightness. GNOME limits itself to 5 levels between zero and max, but the actual hardware usually has at least 10 levels. If the max_brightness file says "10" or more, a finer adjustment will be possible.

$ cd /sys/class/backlight
$ ls
acpi_video0@
$ cd acpi_video0
$ ls
actual_brightness  device@         subsystem@
bl_power           max_brightness  type
brightness         power/          uevent
$ cat max_brightness
10

You can write the desired level to a file named brightness in that directory, but in this example GNOME's own tool is used, which does not require root access.

$ pkexec /usr/sbin/gnome-power-backlight-helper --set-brightness 7

or in more recent versions of GNOME the path changed to:

$ pkexec /usr/lib/gnome-settings-daemon/gsd-backlight-helper --set-brightness 7

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As Linker3000 said, there might be a GNOME Panel applet for controlling the brightness. Right-click the top panel and choose "Add".

The Fn+Key brightness levels are probably setup as a table in the machine's BIOS, but depending on your exact GUI setup (eg: Gnome, KDE etc..), there may be a plugin or widget that you can add to the desktop or status bar that adds a slider for screen brightness. I can't get in front of a Ubuntu machine with a GUI at the moment but I also believe there may be a brightness slider in some of the power management apps - maybe someone else can enlighten...