I know not all of these 'failure' problems during Mountain Lion install are likely to be solved by just one method. But the steps listed below worked for me on a MacBook on which I got the 'failed' error when installing Mountain Lion:
- From screen with 'Installation Failed...' screen, shut down as normal from Apple menu
- Start up again in Safe Mode (hold shift key down until gray apple logo first appears, then release and let startup continue into Safe Mode)
- Log in with an admin-enabled ID in Safe Mode (if you have log-in enabled)
- If you didn't already have a backup of some kind, you might want to take the time to do that while in safe mode and your files are accessible... before doing the next steps.
- Launch Disk Utility from Utilities folder in Applications
- Run 'Verify Disk' and click 'Repair Disk' if any repairable errors are found by the 'Verify Disk' process. This can take a while. (NOTE: In my case, there were NO disk errors found in this process. I ran it a second time to make sure -- same results: NO errors found. So in my case, it was pretty clear the physical disk 'failure' during installation of Mountain Lion was not a true condition.)
- Run 'Verify Permissions' also and click 'Repair Permissions' if any repairable permissions settings need fixing. This can also take a while. (NOTE: I ran the permissions check in addition to the verify-disk because of no errors found in 'Verify Disk' prior step... and I had read in other threads other users' belief that the physical disk itself is not bad, but it's somehow being seen by install process as un-writable which was throwing the error. And in my case there were a number of permissions that needed to be updated / repaired.)
- Shut down normally from Apple menu.
- Start up again normally (NOT in safe mode)
- Let Mountain Lion install process resume and (hopefully) finish, as it did in my case.
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In my particular case, the above steps resulted in the Mountain Lion install starting up again from the default install partition and running without error / failure through completion... and all is working OK now with the new Mountain Lion OS on this MacBook.