How to determine if a black-box is polynomial or exponential
Theoretically speaking, this is impossible to accomplish, basically because "polynomial" and "exponential" are asymptotic concepts, and no prefix of the data guarantees anything about the behavior at infinity.

Practically speaking, you can try to compute $t(n)^{1/n}$ and so if it approaches a constant bounded away from 1. If so, it is exponential. To test whether it's polynomial, you see whether $\log t/\log n$ approaches a constant. These are only rough tests, of course, but they could be useful in practice.