By 'state', I mean a full snapshot of the system, sufficient that you could continue computing from that point. E.g. for a universal Turing machine, this would be a description of the rules, the current tape contents, the current head location, and the state (in the classic TM sense of the word).
This is pretty clearly true in some cases. For example, if I run a program and it tells me the answer is 47, there are infinitely many initial configurations that would terminate there.
This is presuming we aren't doing immaculate bookkeeping and packaging it as part of the state; for a reversible system, this is not so clear. I'm not familiar enough with them to know whether there is typically a limit on the number of unique states through which you can run in reverse. (Would welcome a comment answering this if somebody knows offhand.)
At any rate, I assert there are many systems where all states have infinitely many potential earlier states, but it also seems plausible that is is not a universal property and that there are some easy counterexamples. If someone can provide such a counterexample, or confirm that none exist, I'll consider this answered.
Edit
To be more clear, I'm interested in reachable states. The spirit of the question is essentially this:
For a given Turing-complete system, if you give it an appropriate configuration (program and/or input) such that, after some finite number of steps, it reaches state S, are there going to be infinitely many other configurations that would also reach state S? Does this hold across all possible Turing-complete systems?