The Zeus Machine
It's possible for infinitely many states to take place in a finite time period, as proven by the conceptual possibility of a "Zeus Machine" (see expanded text).
A Zeus machine is an automaton that works faster and faster with each computation it performs.
For example, in listing all natural numbers (an infinite list) the first number is listed in 1/2 second, the second number in 1/4 second and the third in 1/16 second and so on—so that after one second has passed an infinite list has been compiled.
Selmer Bringsjord, 1992.
Note: The mention of a machine that can be in an infinite number of states in a finite amount of time doesn't contradict the earlier claim that machines lack infinite capacity, because a Zeus machine is not a Turing machine or an automaton, but rather an imaginary machine designed to prove a conceptual point.