Ok there are a couple of things that you can consider
The most common is console coma which is where the video region of the onboard eeprom is corrupt. This is relatively easy to fix using a mod chip and eeprom image. It is when you get sound but no video and a Green LED on the front panel.
For more information look up "console coma"
Another option that we dont see often but have done is where one of the timing crystals on the board breaks down.
This is most likely in your case as the timing function that is supplys is inconsistant - ie sometimes it works (you get the timing just right) and some times it doesnt.
The easiest way to determine if the crystal is broken (honestly) is to listen to it !
Take the HDD and DVD out of the case along with all the leads
boot the machine and there is a 27 Mhz (if my memory serves) crystal about 3/4 towards the back of the machine on the left hand side. It looks like a metal oval medicine pill.
If this is making a squeeking noise then it needs replacing - it can be quite quiet though. Oh and the easiest place to get another is literally another XBox - but Component companies will certainly have one.
Like I said - without this timing crystal working - the machine wont boot or will only occasionally boot.
Hope this helps
