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Retrieving Old Timemachine Files
So I am trying to retrieve a file from an old time machine backup from my old computer. However, I keep getting a message on all the folders that I do not have permission to view them (they also have the little "do not enter/NO" symbol). The problem is that that time machine believes those backups were created by someone else. Problem is, I do not have my old computer anymore

The Solution
Go to the folders you want to open in the finder for the old computer backups, and hit command+i for more info.

Click on the lock on the bottom right to unlock it.
Click on the + button on the bottom left to add a new user, then add yourself.

Note
You may have to wait awhile before you appear in the list of people who can read files
Then click on the cog at the bottom and select, "Apply to enclosed items"

Achknowledge the pop-up and wait for the changes to be applied

You can now read the entire contents of the directory and copy them back to your current environment as required