Can't download Mountain Lion from app store on a Mac already running Mountain Lion?
Solution 1:
I would head over to http://support.apple.com/kb/DL1433 and follow the instructions for making a USB Recovery drive.
Also note if you hose your machine you can boot with Command-R to enter into recovery mode. Even if you hose your recovery partition, if you have an active Wifi connection, Apple will pull down the OS from the cloud.
Solution 2:
An alternative method to the one described by Matt is to use Lion Diskmaker. Despite its name Lion Disk Maker is an application programmed with AppleScript that you can use with Mac OS X 10.6, 10.7 and OS X 10.8 to burn a DVD or build a bootable drive from Mac OS X Lion or OS X Mountain Lion Installation program.
Solution 3:
If you have a newly-purchased Mac, it probably runs a version of OS X Mountain Lion that came out after 10.8.2 was already released. What this means is your Mac actually requires a version of Mountain Lion that is newer than what is available on the Mac App Store. The one on the App Store does not have all the requisite drivers bundled in to talk to all of your hardware.
In the future, once 10.8.3 is released, you will be able to successfully download that from the App Store, and use a tool like Lion DiskMaker to make a bootable copy.