Things to check when my laptop shuts itself down after being disconnected from the power:
- Try re-seating the battery for starters. Physically remove it and plug it back in.
- Can the battery hold a charge? More difficult to diagnose but look under the a desktop applet to see if the battery is fully charged.
- Is the laptop configured to shutdown when it gets to a certain threshold of battery charge left? Try changing this value to see if the laptop can get past being unplugged for any length of time. This will tell you that the issue is the battery.
For #3 take a look at dconf-editor to see if the laptop is setup to shutdown when the battery is critical.

You can also query these options from the command line using the gsettings command. Not sure if this command is available on all distros. I'm testing it on Ubuntu at the moment:
$ gsettings list-recursively org.gnome.settings-daemon.plugins.power
org.gnome.settings-daemon.plugins.power active true
org.gnome.settings-daemon.plugins.power button-hibernate 'hibernate'
org.gnome.settings-daemon.plugins.power button-power 'interactive'
org.gnome.settings-daemon.plugins.power button-sleep 'suspend'
org.gnome.settings-daemon.plugins.power button-suspend 'suspend'
org.gnome.settings-daemon.plugins.power critical-battery-action 'suspend'
org.gnome.settings-daemon.plugins.power idle-brightness 30
org.gnome.settings-daemon.plugins.power idle-dim-ac false
org.gnome.settings-daemon.plugins.power idle-dim-battery true
org.gnome.settings-daemon.plugins.power idle-dim-time 30
....
dmesg's output. Is there anything possibly-related?