If you get interrupted, anyone can stop the bleeding with a makeshift tourniquet. You'd end up doing quite a bit of damage to nerves and tendons as well. If you're interrupted quickly enough they may be able to repair some of that damage, but not all of it. It technically is possible to die by cutting alone but it really only has like a 6% chance of fatality if I'm remembering correctly. Then again, there's probably people that tried it, and failed, that never even made it into those kinds of statistics so it may even be lower than that.
The pain is also a factor that a lot of people can't get past unless you're a long-term self-harmer.
But if you're really set on it, I recommend you do quite a bit of research on where to cut on the forearm/arm. I personally wouldn't want to have to keep digging around to locate the arteries. It'd probably also be better to hit the Brachial Artery in the arm over the Ulnar or Radial in the wrist but that's just my personal opinion and I, again, recommend more research into this method. Or even look into Blood-Letting?
As for the combo of overdosing with cutting. It depends on what you use.
Non-prescription/ OTC drug overdose alone only has an exceptionally low rate of fatality and it's not even usually the initial overdose that kills you, but the Liver or Kidney damage as a result of said overdose.
Prescription drugs also has a high chance of failure unless you really research it. I almost died after an overdose on Lithium-Carb because there's such a low threshold from therapeutic dosages to toxic doses- I was unfortunately found in time and spent eight absolutely miserable days in the ICU before I was carted off to the psych ward. I believe, if I'm remembering it right, I took roughly 350 pills of 300mgs of Lithium Carbonate- I can't remember if it was extended release or not as it's been a few years now.
So death by cutting with an overdose is technically possible but there's still a high risk of survival. Again, more research would need to be done on this if you're dead set on it as a method.
I hope this helps you some and if anything doesn't make sense, please feel free to ask me, or anyone else, questions. It's currently extremely late here, or early, and I'm rather sleep deprived.