The wake on LAN feature can be affected by any peripheral connected to the motherboard, a video card, power supply, hard drive, optical drive and even RAM memory could be the culprit of the system not waking up.
I recommend you to test with a different power supply and to make sure that the motherboard is using the latest BIOS available.
To properly assist you, as this is a desktop motherboard forum, the model number of the motherboard will be appreciated.
It is a Dell Optiplex 7010, but I was more concerned with the network card settings. Wake on LAN works immediately after the computer is shutdown and for a couple of hours after, but at some point several hours later the Wake on LAN will not work. This made me think it was a network card power management setting, but I cannot seem to narrow it down further.