Answer:  In general, yes. When an employee is on a performance improvement plan (PIP), and their performance has not improved and has, in fact, gotten worse, it is perfectly reasonable to cut the timeframe of the PIP short and move forward with further disciplinary action, including termination. Unless it’s written to say otherwise—and it absolutely shouldn’t be—a PIP is not a guarantee of employment for the duration of the plan. It shouldn’t alter the at-will employment relationship.
Just be sure that you are following historical practices if you have had similar situations in the past. The most important thing is to remain consistent. Document—and tell the employee—the reason why the PIP was cut short, listing each policy violation or performance issue individually, in case you are asked to provide context at a later date.