I've found that it is 4 hours from the time of your first hack. And it is triggered after you have successfully hacked the same portal four times in four hours (e.g. the fifth hack attempt will result in the "Portal burned out!" message.
What I haven't determined is if this is a rolling four hour window.
For example, you hack Portal A at 08:00, then again later at 11:00, 11:05 and 11:10. If you attempt to hack Portal A again at 11:15, you'll get the "Portal burned out!" message. But, if you try again at 12:01, your hack will be successful (because four hours have elapsed since your first hack at 08:00.)
Now, if you attempt a sixth hack at 12:06, will it succeed because your "clock" has been reset at 12:00 and you now get a full four hacks in the next four hours? Or will it fail, because you have hacked the portal successfully four times since 11:00?
Further testing is needed.