Is it common that you can't finish the exam questions during PhD study? I'm a first year student and twice now I have been unable to finish all the problems during the exam, which had never happened before I became a PhD student. And to be fair, even if I had been writing without stop, I might still have failed to finish the exam questions. Is this common? I believe my professors are trying to push us beyond the limit, but it is frustrating.

P.S. I'm not complaining, I just try to cope with it with right attitude.
This is completely dependent on the course and instructor.

Yes, I have taken exams where it definitely wasn't possible for anyone to finish. The professors either gave us more time or only graded the problems we attempted.

Talk to the instructor about your concerns!