Yes. Motherboards on desktops get several voltages from the PSU unit via the big power connector (plus an aux 4 pin 12V connector) but the power for the memory, CPU, and motherboard controller come from on-board siwtching power supplies which themselves are powered by the 3.3V/5V/12V lines from the PSU.
Because of the very high current CPU's use, and the sensitivity to voltage levels (for both the memory and CPU) these on-board power supply converters are usually quite complex, and always have fault protection to do their best to prevent current surges and over voltage. Each on board power supply always has a connection to the motherboard controller for checking the status - this may be a simple 'power ok' line, or a data connection where bytes are exchanged. So, if one of the switching transistors that one of these power supplies blows (MOSFETs close circuit) the converter cannot hold regulation, flags up an error, and the motherboard controller powers it down to prevent further damage to the CPU/memory. The happens all before the bios is even looked at - its pretty much pure hardware. And will halt all signs of operation, including the CPU fan. Any fans connected to the motherboard *may* still spin, because they will get the 12V from the PSU, but their speed is controlled by the motherboard controller. I would say, if they don't spin at all then the mobo has detected a fault, and shutdown.
Even if these converters are working perfectly, if they are powering anything that has shorted (example: bad RAM) - then too much current will be drawn, the converter detects this, and the above process happens. In this case, no damage is done, its just the mobo refusing to do anything because a faulty component is installed.
I've been working on laptop motherboards recently, which is a slightly different ball game, because most of the components are integrated, where-as in desktop's, where space is of no concern, its modular. So bare with me, as I haven't worked on a desktop in months.
Unless you have access to another system that you can use to test individual parts, such as: Drives (SATA), PCI cards, memory, even CPU etc.. then removing everything except the absolute minimum is a good start. If your motherboard has on board graphics - which I'm pretty sure almost all do these days, then remove all PCI cards, including the graphics card. Remove all SATA devices. Leaving only the CPU and
one stick of RAM. Your monitor can be plugged into the VGA (or HDMI) connector on the motherboard.
So, with no drives, there's no OS, and no drivers, this will check just the motherboard, CPU and ram. If it fails to power up at all, change the RAM stick to the other one you have. I would be prudent to test the memory in another machine to completely eliminate that as a problem. Leaving only the motherboard, and CPU. On power up, it should happily get to the BIOS load up screen, then complain that there is no operating system found. Adding parts back one by one until things start to go less smoothly, and you should have found whats giving you greif. I would start with hard drive, then graphics card, then other PCI cards, then any other RAM.
As its an intermittent problem, and one that happens before the OS loads, it is likely to be a power issue (think I've said that before..). The above operation is basic, time consuming, but is a good methodical way of finding out whats up, and something I wish I had done more often as it would have saved me much time in trying to work out whats wrong.
When I said 'parts', I meant, pretty much everything

Once you've narrowed it down to the a single 'major' part (like motherboard, a PCI card etc...) then if its the motherboard, you can probe that for power problems. Apologies for giving vague advice, as I said, my head has been full of laptop schematics and datasheets, so I get lost in the 'detail' rather than the bigger picture.
Yep, I hate computers