The pseudorandom sequence is something like:
new_state = (old_state * C1 + C2) modulo N
where C1 and N are probably prime, or at least prime relative to each other, so that the sequence would travel through every possibility before repeating. Presumably nowadays the randomizing function is more complicated.
I also remember a story about people cracking an online poker site because they worked out the random number generator sequence and thus had a good idea of which cards the other players were likely to have. The headline had me thinking it was a made-up story, but when digging into it a bit deeper, it actually seemed plausible.
Anyway, I am off to dump a few random numbers, see how we go. 16-bit sequences would be no problem, 32-bit would be doable with effort, 64-bit would be... interesting.