I've cut myself (not suicidally, just cutting) for 20+ years. I've done it severely enough to warrant stitches countless times. I've hit veins on accident, but I've never hit an artery. Arteries are DEEP. I imagine you'd have to get one to bleed to death though, veins don't have a pulse pumping blood out faster than it can clot. I've had a ton of experience slicing myself up and you simply can't bleed enough from veins only.
There are some conditions that may make it possible. if you were on blood thinning medication it may be easier. OTC pain meds like aspirin will thin your blood a little. Being fairly drunk also thins blood. If you had a blood or bone marrow disorder that left you with very low platelets exsanguination may work as a method. For the most part very few suicides are completed by bleeding to death, despite the prevalence of wrist cutting as a method in pop culture. It's just too difficult to get to arteries for most.
I honestly don't think it seems like a terribly painful method, but Im biased as a cutter. Arteries are much more painful to hit than just flesh and veins; I know this from working in healthcare, arterial blood draws are more painful than regular venous ones. But aside from pain at the wound it would be a fairly easy death you would get lethargic and pass out before dying.
lastly, there's a very high chance of changing your mind and time enough to self rescue with this method. A severe wound and the sight of blood will trigger SI for most people. The shock and subsequent endorphine rush will change your state of mind pretty rapidly. You'll may feel a whole lot better about being alive after you cut. That's why it's so habit forming!