Predicting the next number output by a random number generator
It is a well-known fact concerning (cryptographically secure) pseudorandom generators (PRGs) that pseudorandomness is equivalent to next-bit unpredictability. That is, distinguishing the output of a PRG from the uniform distribution is just as hard as predicting its next bit of output (when the PRG works on a uniformly distributed seed). This goes back to a result by Yao.

Further reading:


Yao's test
Next-bit test
Pseudorandomness




If we are talking about standard (i.e., not necessarily cryptographically secure) random number generators (RNGs), then things are a bit different. If the RNG simply has to fulfill a set of heuristics, then it can be the case that it does so despite being (almost) fully predictable. (As an extreme example, consider the rather naive heuristic $\lim_{n \to \infty} E(|G(1^n)|_0) = \lim_{n \to \infty} E(|G(1^n)|_1) = \frac{1}{2}$, that is, the string $G(1^n)$ tends to have just as many zeroes as ones and the "generator" $G(1^n) = 0^{\lfloor \frac{n}{2} \rfloor} 1^{\lceil \frac{n}{2} \rceil}$.) 

Breaking such an RNG would involve statistical tests and ad hoc methods; your chances of success depend on what the heuristics are and how much computational power and RNG data you have available. Without having detailed these things, it cannot really be said whether it is feasible to break it or not.