However, simple generators (unlike the Mersenne Twister) are really deterministic, so if you observe the same number a second time,

A generator of the integers 0 .. 2, can generate those 3 values in 6 different sequences:

a: 0 1 2
b: 0 2 1
c: 1 0 2
d: 1 2 0
e: 2 0 1
f: 2 1 0
[download]

And those 6 (sub)sequences can be output in 720 permutations before repetition is required:

a b c d e f
a b c d f e
a b c e d f
a b c e f d
a b c f d e
a b c f e d
a b d c e f
a b d c f e
...
f e c d b a
f e d a b c
f e d a c b
f e d b a c
f e d b c a
f e d c a b
f e d c b a
[download]

That's not strictly true, because it is likely that some 18-digit overlaps between two subsequences will replicate an earlier subsequence.

But then, the last 2 digits of subsequence a, and the first digit of subsequence b; form an out-of-sync repetition of subsequence d. and given that MT can only produce 2**32 values; similar, out-of-sync subsequence repetitions have to have occurred many times to arrive at the 2**19937-1 periodicity.

So, repetition of a single value, or even a single subsequence is not an indicator that periodicity has been reached.