The original Nondeterministic Time Hierarchy Theorem is due to Cook (the link is to S. Cook, A hierarchy for nondeterministic time complexity, JCSS 7 343–353, 1973). The theorem states that for any real numbers $r_1$ and $r_2$, if $1 \le r_1 \lt r_2$ then NTIME($n^{r_1}$) is strictly contained in NTIME($n^{r_2}$).
One key part of the proof uses (an unspecified) diagonalization to construct a separating language from the elements of the smaller class. Not only is this a nonconstructive argument, but languages obtained by diagonalization usually provide no insight other than the separation itself.
If we want to understand the structure of the NTIME hierarchy, the following question probably needs to be answered:
Is there a natural language in NTIME($n^{k+1}$) but not in NTIME($n^k$)?
One candidate might be k-ISOLATED SAT, which requires finding a solution to a CNF formula with no other solutions within Hamming distance k. However, proving the lower bound seems is tricky, as usual. It is obvious that checking a Hamming k-ball is clear of potential solutions "should" require $\Omega(n^k)$ different assignments to be checked, but this is by no means easy to prove. (Note: Ryan Williams points out this lower bound for $k$-ISOLATED SAT would actually prove P ≠ NP, so this problem does not seem to be the right candidate.)
Note that the theorem holds unconditionally, regardless of unproved separations such as P vs. NP. An affirmative answer to this question would therefore not resolve P vs. NP, unless it has additional properties like $k$-ISOLATED SAT above. A natural separation of NTIME would perhaps help to illuminate part of the "difficult" behaviour of NP, the part which derives its difficulty from an infinite ascending sequence of hardness.
Since lower bounds are hard, I will accept as an answer natural languages for which we may have a good reason to believe a lower bound, even though there may not yet be a proof. For instance, if this question had been about DTIME, then I would have accepted $f(k)$-CLIQUE, for a non-decreasing function $f(x) \in \Theta(x)$, as a natural language that probably provides the required separations, based on Razborov's and Rossman's circuit lower bounds and the $n^{1-\epsilon}$-inapproximability of CLIQUE.
(Edited to address Kaveh's comment and Ryan's answer.)