http://serverfault.com/questions/463604/unable-to-install-net-3-5-on-windows-server-2012

This behavior can also be caused by a system administrator who configures the computer to use Windows Server Update Services (WSUS) instead of the Microsoft Windows Update server for servicing.

http://support.microsoft.com/kb/2734782

This worked for me. Windows has to download the 3.5 installation files, but the server is configured not to use Windows Update (common for managed servers), but WSUS. The above article describes how to fix this. In a nutshell:

Start the Local Group Policy Editor or Group Policy Management Console.
Expand Computer Configuration, expand Administrative Templates, and then select System.
Open the Specify settings for optional component installation and component repair Group Policy setting, and then select Enabled.
Select the Contact Windows Update directly to download repair content instead of Windows Server Update Services (WSUS) check box.

Make sure Windows Updates Service is set to Manual or Automatic to apply this fix. Our default images are set to disabled, and the issue continued until that seemingly obvious change was made.