Narrowed down the root cause to a failed power circuit on one of the two power connectors on my graphics card. Pretty weird how it would cause the PC to not power on--looks like it was triggering a protection mechanism on the PSU.
Was worried the 2 year warranty was just about over, but it turned out it's got 3 years, so I should be fine. Was considering upgrading to a GTX 970 or something, but I guess I'll hold off for a while longer.