Is it possible it's a wake from sleep issue? Check your settings in system preferences>energy savings. If it’s set to sleep in 15 min. and then is having trouble waking that would explain it. On the other hand, it might be a power supply or some component that when it heats up will power off. Check to see if all fans are working. The not completing boot up could be bad ram or a corrupt OS or a hardware issue.
Try booting from an external drive and see if problem is fixed.
Remove any 3rd party devices plugged into Pro and see if problem is fixed.
Remove extra ram chips and go back to original ram. See if problem is fixed.
Repair permissions. Repair disk. Reset p-ram.
See if problem is fixed.
Run hardware check app that came with Mac Pro.
Reinstall OS and see if problem is fixed.
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