This doesn't come up very often, but you should try the following:
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In most cases, the downloaded dmg file is actually corrupt or had an error downloading. If possible, try downloading the dmg again, turning off any download assistant plug-ins you may have. You can try downloading the file in a different browser as well. Or if you don't need to be logged in to the site to download the file and you want to be fancy, you can try curl -O url in Terminal to download the file. (There's an example of that in my screenshot below.)
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Reboot your Mac if you haven't already tried that. Apparently there is an issue sometimes after opening too many dmg files, that is fixed with a reboot.
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Try mounting the DMG on the command line in Terminal. We will at least get some sort of useful error message to go on if it still fails:
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Open Terminal: In Spotlight, the search magnifying glass at the upper right corner of your screen, search for Terminal, and press enter to open the Terminal app.
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Type hdiutil attach into the terminal. Don't press enter yet.
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Drag the dmg file from your Finder window onto the Terminal window and let go. This will fill in the location of the dmg file into your Terminal window.
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Press enter.
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Think about if you have any kind of security policies on this machine to prevent writing to external drives (thumb drives, optical drives, etc). I haven't seen this one in action, but I read about this being a possibility while researching the issue.
For example, there may be a security policy that prevents mounting a thumb drive, CD drive, other external drive on your machine that is affecting the dmg.
