One thing to consider, I had an issue on several pc's at work a few years back, turned out there was a bug in the chipset that caused certain graphics cards to fail to initialise when the pc was first switched on. (Not gigabyte btw)
Nforce chipset and Geforce cards the remedy was to press reset , or in extreme cases power off and on again,
this problem was intermittent but of course for the users rather annoying.
This turned out to be some problem on the AGP bus on the old nforce chipset
I am wondering if there is some kind of bug on your chipset that maybe a bios update will help.
of course we don't have your specs, so cannot comment on what may be wrong.
on another note, some chipsets regardless of motherboard manufacturer can be very fussy with ram.
So some nice Kingston HyperX 1600mhz ram that worked flawlessly on a 790X board causes random BSOD on a 790FX board
not to mention the budget 770 board from MSI that refused to work with any budget Ram, meaning that the saving made on the board was lost when expensive ram had to be fitted!
So try chaning the Ram or VGA Card if you can
Good Luck