This happened to me a while back. It would get to the login screen, and I could type in a username/password, but at some point it would go blank, and restart X.
It eventually went away when I upgraded to a different video driver, and inspite looking for it, I couldn't find anyone else talking about the same symptoms.
I suggest you try a few things, which you may already have done. Firstly try pressing Ctrl+Alt+F9 to see if you can terminate X and get back to the text screen. Secondly try ssh-ing into the machine when it's gone black, see if it's still running, or if it has kernel dumped.
In my case it would come back, after a restart, but it was pretty much guaranteed to die when it was first started, then as I said, it started working perfectly, before ultimately, my GPU died. My laptop is an XPS, and apparently its a common problem that they run hot, over time, and the GPU comes unsoldered. The blank screens could a sign of something crashing intermittently.
lspci | grep VGA
? It spits out details about your graphics card. – willem.hill Apr 13 '16 at 02:43