Does "does not achieve x and y" equal "achieve neither x nor y"?

I believe that "neither x nor y" is least ambiguous way to express it concisely.

However, an even clearer way to say it would be to give up on conciseness:

If Tenant fails to X and also fails to Y, Tenant may terminate the lease.

If you had more than 2 or 3 conditions, you could express them as a list, something like:

If Tenant fails to achieve each of the following conditions, Tenant may terminate the lease:

1. A
2. B
3. ...

However, as someone mentioned in the comments, for legal documents your best bet is to consult a lawyer. There's often specific terminology that has become customary. Leases like yours are probably not unique, and there's almost certainly precedent for language that should be used.

From a logical perspective, the answer is no. They are not equivalent.

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Does not achieve X and Y.

This statement is true if any of the following results occur:

1. Only X is achieved.
2. Only Y is achieved.
3. Neither X nor Y is achieved.

But here is your second statement:

Achieves neither X nor Y.

This is only true in the third result listed above.

If the first or second result occurs, then does not achieve X and Y will be true—but achieves neither X nor Y will be false.

Therefore, your two statements are not logically equivalent.

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For further clarification, in order for the logical nor to evaluate to "truth," the conditions on both its sides must evaluate to "false."

Or (as per Wikipedia):

In boolean logic, logical nor or joint denial is a truth-functional operator which produces a result that is the negation of logical or. That is, a sentence of the form (p NOR q) is true precisely when neither p nor q is true—i.e. when both of p and q are false.

Logicians like things to be clear and succinct, but the English language was not designed by logicians. It is still being developed as it has been for the last 1000 years by a committee of over 1,000,000,000 people, most of whom have no understanding of logic. If you want to be clear, you may have to forego a little succinctness and add a couple of words:

1. If Tenant does not achieve x and does not achieve y then Tenant may terminate the lease.