Not a reasonable idea, no. It would be possible, technically, but it's also possible to die by ingesting enough plastic debris, or enough pennies or too much water.
For reference, losing 40% of your blood is usually fatal, and the average adult has 4.5 to 5.5 liters of blood, which would mean about 2 liters (half a gallon or so) of blood loss. Donating blood involves giving between 300 and 500 mls of blood, and if you've ever given blood, you know it actually takes a while to get that amount out of you with an IV. You'd need to extract at least 4 times more to deliberately die, and the more you extract, the slower it goes, because less blood means less blood pressure and a slower flow rate.
One of the more interesting case reports I've read about suicidal exsanguination was a doctor who injected himself with lidocaine (a local anesthesic) near the femoral artery, and then cut the femoral artery with a sharp blade... I've personally considered this method - basically painless, minimal risk of serious injury in the event of failure, likely to be highly effective.