There are problems that cannot be solved in polynomial time on a nondeterministic machine. Indeed, there are problems that cannot be solved on any Turing machine, such as the halting problem.

We know from the nondeterministic version of the time hierarchy theorem that there are things that can be done in exponential time on a nondeterministic TM that cannot be done in polynomial time. That is, $\mathrm{NP}\subsetneq\mathrm{NEXP}$.

Wikipedia gives several examples of $\mathrm{NEXP}$-complete problems: these are examples of problems that are provably not in $\mathrm{NP}$. Determining the existence of winning strategies in some games is also $\mathrm{NEXP}$-complete.

There are problems that cannot be solved in polynomial time on a nondeterministic machine. Indeed, there are problems that cannot be solved on any Turing machine, such as the halting problem.

We know from the nondeterministic version of the time hierarchy theorem that there are things that can be done in exponential time on a nondeterministic TM that cannot be done in polynomial time. That is, $\mathrm{NP}\subsetneq\mathrm{NEXP}$.

Wikipedia gives several examples of $\mathrm{NEXP}$-complete problems: these are examples of problems that are provably not in $\mathrm{NP}$.