Solid solution, David.
One small error: x cannot equal 3/4 (since z must be an integer)

Cheers,
Brent

Mo2men has raised a very interesting point. It has caused me to change the official answer to D, unless I'm missing something.

I have also upgraded the difficulty level to 800!!!!!

Given: x - y - z < 0

Target question: Is z > 1

(1) x - y > 1 - z

From statement 1 (and the given inequality), we learn that z > 1/2. So, we might (incorrectly) conclude that it could be the case that z = 1 or z = 2, in which case, we get different answers to the target question.
HOWEVER, if we try to come up with values for x, y and z that demonstrate this, we find that we have a problem.

If z = 1, then we can plug this value into our two inequalities.
For the statement 1 inequality, we get x - y > 1 - 1
Simplify to get: x - y > 0

For the given inequality, we get x - y - 1 < 0
Simplify to get: x - y < 1

When we combine the two inequalities, we get: 0 < x - y < 1
In other words, the difference between x and y is a fractional value BETWEEN 0 and 1.
This is IMPOSSIBLE, since it's given that x and y are integers.
So, it cannot be the case that z = 1

Since we already know that z > 1/2, we CAN conclude that it's possible that z = 2, z = 3, z = 4, etc.
For example, consider these situations:
Case a: x = 0, y = 0 and z = 2. In this case, z IS greater than 1
Case b: x = 0, y = 0 and z = 3. In this case, z IS greater than 1
Case c: x = 0, y = 0 and z = 4. In this case, z IS greater than 1
Case d: x = 0, y = 0 and z = 5. In this case, z IS greater than 1
etc..

So, it turns out that the correct answer is actually D (both statements are sufficient)

Sorry for not knowing the correct answer when I first posted the question. It's even harder than I first imagined!!

Cheers,
Brent