Could you simulate a Turing machine by a sequence of Turing machines each with strictly fewer states than the simulated machine? By a sequence of Turing machines I mean this: the first machine is given the input tape, then executes until it accepts, then the contents of the tape are given to the next machine in the sequence as input, and so on until the final machine accepts. If any machine in the sequence rejects, the simulated machine would reject, and if any in the sequence would fail to halt then the simulated machine would fail to halt.
If this is possible, the more interesting question would be whether there is a bound on the number of machines needed to simulate a machine with more states. The motivation behind this question is that Busy beaver numbers are known for very small machines, so I was curious about whether machines with many states could be (reasonably) reduced to machines with fewer states.
It seems to me that there must be some error in reasoning or understanding if the number of machines needed to simulate another could be bounded, because that would imply any machine could have a bound on steps needed before halting, but I am unsure where that disconnect is.