Can a diode fail to open position?
A diode fails closed due to overvoltage. This is called punch-through. This is is the principle used in ESD diodes. If they can't handle the voltage the PN juntion fails and short to ground, protecting any circuitry after them. On the image you see a small black dot where the voltage went through the junction. eg:




*Second picture added that shows this better on a BJT.

A diode typically fails to open happens due to over current. This is called metallization burnout and can occur from things like EOS (Electrical Over Stress). Image shown bellow. Over current causes excessive heating and literally burns the metal away. As mentioned above this is easy to demonstrate on LEDs as their current carrying capability is much lower than rectifier didoes.