Electronic – Is the zener diode labeled wrong, or am i seeing a strange failure mode

The way you've drawn it the zener diode clamps the power supply reducing the 34V to 12V. What's even worse is that there's no current limiting and the diode will be destroyed.

The zener diode needs a series resistor to take care of the voltage difference between the 34V and the 12V.

To calculate the resistor value you need to know how much current the zener diode needs and how much current should go into the load (the opamp in your drawing). Zener diodes are often specified at the rather high current of 50mA, though more modern types will work properly at current less than 1mA. (This one only needs 50\$\mu\$A.) Let's suppose it needs 10mA and that the opamp will also consume 10mA, that's a total of 20mA. That's the current through the resistor. Since we already know the voltage across the resistor Ohm's Law will give us the value:

\$ R = \dfrac{\Delta V}{I} = \dfrac{34V - 12V}{20mA} = 1100\Omega \$

If you check the power dissipated in the resistor (something you always should do) you'll see that this is 0.44W, so use a 0.5W power resistor.

edit
My original answer talked about 50mA zener current. That's very high, and I also realized that at 50mA a 12V/400mA zener would operate out of spec, as also jippie pointed out. It appears that the high current is only used for lower voltages (76mA for the 3.3V 1N4728A), and at 12V the datasheet talks about a safer value of 20mA. It's OK to use a lower current, though load and line regulation will be worse.

That looks entirely normal to me. 3.3V zeners have a very soft knee and are unsuitable for this purpose. The zener voltage is guaranteed to be between 3.1V and 3.5V at 5mA test current, which agrees with your number with an 8V input (3.2 \$\le\$ 3.4 \$\le\$ 3.5V) with zener current approximately 4.6mA. In other words, it's behaving exactly like a 3.3V zener diode, which in this case is not good at all.

A diode to a clamp will do better, but without more information it's hard to make recommendations. A TL431 with a 1mA current (from a 5V circuit) and a diode to your ADC input should work, but you'll have to keep the clamp voltage to maybe 3V, and some current still will flow into the ADC input. A R-R IO op-amp will clamp it more positively.

If your actual input voltage never exceeds 1 volt, the datasheet I linked above guarantees that the current at 25°C will not exceed 2uA, so your error will be less than 2mV, but at 125°C it could be as bad as 40mV error. Chances are you want to use more than the bottom 1/3 of your ADC range.

Edit:-

"Zener" diodes of greater than about 5.6V are actually avalanche diodes, and they have a much sharper knee and a positive tempco. Here is a set of curves from a totally different family of small zener diodes (lifted from a 20-year-old paper Toshiba databook). As you can see, at about 8.2V and above, they are really, really good, with little voltage change from 1uA to 10mA. The "3.3V" one changes from about 1.25V to about 3V. This is a consequence of the physics involved, and you'll find some zeners shift the curves up and down a bit, but the shapes will be similar. Active circuits are required to do much better.

Unfortunately, for some reason, this kind of info is often omitted from modern datasheets, even though it was considered important enough to murder trees for in the old days.